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ROBUST TRANSITIVITY

FOR ENDOMORPHISMS

Cristina Lizana Araneda

A thesis submitted for the degree of

Doctor of Phylosophy

in Mathematics

Advisor: Enrique R. Pujals.

Rio de Janeiro – Brazil

June 2010

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Abstract. The main goal of this work is to give some necessary and some suﬃ-

cient conditions for en domorphisms on compact manifolds without boundary to

be robustly transitive. More concretely, under what conditions a di ﬀerentiable

map, not necessarily invertible, having a de nse orbit, veriﬁ e s that a suﬃci e ntly

close perturbed map also exhibits a dense orbit.

In the case of robustly transitive diﬀeomorphisms is known that a nece s sary

cond i tion is that the tangent bundle admits a dominated splitti ng. For the

case of endomorph i s ms, that is no lo nger true. In consequence, conditions

that g uarantee robustness for transitive endomorphisms cannot depend on the

existence of decomposition of the tangent bundle.

For local diﬀeomorphisms, we show that a necessa ry condi tion for robust transi-

tivity is to be volume expanding. Although volume expanding is not a suﬃcient

cond i tion to ha ve endomorphis ms robustly transitive. Because of this, we must

ask for more hypothesis that guarantee robustness. Indeed the additional hy-

pothesis that we as k is: given an y arc in a certain region with a large enough

diameter to have a point that its f uture orbit remain s in the expanding reg i on,

which implies the existence of a locally maxima l expanding invariant set for the

original system that intersects every arc big enough.

F´acilmente aceptamos la realidad,

acaso porque intuimos que nada es real.

J.L. Bor ges

Acknowledgement

I wo uld like to acknowledge:

To my advisor Enrique Pujals for accepting me as his student. I really enjoyed our

long talks about mathematics and many other subjects, I am grateful for his patience,

encouragemen t and friendship.

To the members of the jury: Jacob Palis, Marcelo Viana, Flavio Abdenur and Andr´es

Koropecki. Specia lly to Andr´es for many good advices and tips to improve this thesis a nd

for useful discussions abo ut math.

To IMPA for the very good e nvironment and facilities.

To TWAS and CNPq for the ﬁnan cial support that make it possible.

To Universidad de Los Andes for the partial ﬁnancial support.

To Mike Shub, Chri s tian Bonatti and Martin Sambarino for useful conversations during

the “Interna tional Congress on Dynamical Systems” in B´uzios last February.

To Leonardo Mora for his encoura g ement and good advice.

To my family: Marcos, Elena, Isa and Dey, for their love an d support in this long

journey.

To my colleagues for many good discussions, they listened me when this was still a

project. Specially I would li ke to thanks Panch o, Guarino, D´avalos, Carlos and Yuri.

To many good friends that make the life easier and happier during my stay in Rio.

Contents

Introduction iii

1 Main Result 1

§ 1.1 Volume expanding endomorphisms without invariant splitting . . . . . . . 1

§ 1.1.1 The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

§ 1.1.2 Sketch of the Proof of Main Theorem . . . . . . . . . . . . . . . . . 10

§ 1.1.3 Existence of a Locally Maximal Set fo r f . . . . . . . . . . . . . . . 11

§ 1.1.4 Continuation of the Locally Maximal Set . . . . . . . . . . . . . . . 14

§ 1.1.5 The Locally Maximal Set “Separates” . . . . . . . . . . . . . . . . . 18

§ 1.1.6 Getting Sets of Large Diameter . . . . . . . . . . . . . . . . . . . . 26

§ 1.1.7 Proo f of The Main Theorem . . . . . . . . . . . . . . . . . . . . . . 28

§ 1.1.8 The Main Theorem Revisited . . . . . . . . . . . . . . . . . . . . . 29

§ 1.2 Volume expanding endomorphisms with invariant splitting . . . . . . . . . 30

§ 1.2.1 Theorem 2: Splitting Version . . . . . . . . . . . . . . . . . . . . . 31

§ 1.2.2 Proo f of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

ii Contents

§ 1.3 Remarks About the Main Theorem and Theorem 2 . . . . . . . . . . . . . 33

2 Existence of a semiconjugation to a linear expanding endomorphism 35

§ 2.1 Dynamical Consequences: Geometrical and Topological . . . . . . . . . . . 35

§ 2.2 The Case f is Isotopic to a Linear Expanding Endomorphism E . . . . . . 36

§ 2.2.1 Existence of the Semiconjugation . . . . . . . . . . . . . . . . . . . 36

§ 2.2.2 Markov Partition and Transitivity . . . . . . . . . . . . . . . . . . . 38

3 Existence of Robust Transitive Endomorphisms 45

§ 3.1 Example 1: Applying Main Theorem . . . . . . . . . . . . . . . . . . . . . 45

§ 3.1.1 Property of Large Arcs . . . . . . . . . . . . . . . . . . . . . . . . . 47

§ 3.1.2 Remarks About Example 1 . . . . . . . . . . . . . . . . . . . . . . . 48

§ 3.2 Example 2: Applying the Main Theorem Revisited . . . . . . . . . . . . . 48

§ 3.2.1 Λ

Separates Large Nice Cylinders . . . . . . . . . . . . . . . . . . 50

§ 3.2.2 Remarks About Example 2 . . . . . . . . . . . . . . . . . . . . . . . 52

§ 3.3 Example 3: Applying Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . 52

§ 3.3.1 Λ

and Λ

Separate Large Vertical Segment s . . . . . . . . . . . . . 54

§ 3.3.2 Remarks About Example 3 . . . . . . . . . . . . . . . . . . . . . . . 55

§ 3.4 Example 4: Applying Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . 56

§ 3.4.1 Λ

Separate Large Unstable Discs . . . . . . . . . . . . . . . . . . . 57

§ 3.4.2 Remarks About Example 4 . . . . . . . . . . . . . . . . . . . . . . . 58

Bibliography 59

Introduction

One goal in dynamics is to look for conditions that guarantee that certain phenomena are

robust under perturbations, that is, some main feature of a dynamical system is shared

by a ll nearby systems. In particular, we are int erested in the hypotheses under which an

endomorphism would be robust transitive.

In the diﬀeomorphism case, there are many examples of robust transitive systems. The

best known is the transitive Anosov diﬀeomorphism; the example given by Shub in T

1971, see for instance [PS06 b]; another example is the Ma˜n´e’s Derived from an Anosov in

in [Ma˜n78]; Bonatti and D´ıaz gave some geometrical construction that induces a robust

transitive system in [BD96]. One of the newest examples is one from Bonatti-Viana that

gives a robust tr ansitive diﬀeomorphism wit h dominated splitting which is not partially

hyperbolic, see [BV00].

On the other hand, any C

−robust transitive diﬀeomorphism exhibits a dominated

splitting. This is not true anymore for endomorphisms (see endomorphisms version of

Bonatti-Viana example, it is explained in example 1 in section § 3.1). Therefore, conditions

that imply robust transitivity cannot hinge on the existence of splitting.

The ﬁrst question that arises is what necessary condition a robust transitive local

diﬀeomorphism has to verify. We show in the ﬁrst chapter that volume expanding would

iii

iv Introduction

be a C

necessary conditio n. However, volume expanding is not a suﬃcient condition

that g uarant ees robust transitivity for a local diﬀeomorphism, for instance a product of an

expanding endomorphism times an irrational rotation is volume expanding and transitive

but not robust transitive. Hence, we have to ask for extra conditions that allow us to

conclude the robustness, more precisely, we need a property over the initia l system that

would be robust. As we said before this pr operty cannot depend o n the existence of any

type of splitting. In fact, t he hypothesis that we require is to ask for any arc of large

enough diameter t o have a point such that its forward iterates remain in the expanding

region, implying the existence of an invariant expanding locally maximal set for the initial

system that intersects every large arc. Therefore, that topological property persists under

perturbations for a certain class of arc.

Instead of transitivity we may ask for the density of the pre-orbit of any point. This

hypothesis implies transitivity, but we do not know if the reciprocal holds. But the fact

of having just one point which pre-orbit is dense is not enough to conclude transitivity.

We must note that if the initial system ver iﬁes the density of every pre-orbit, then t he

perturbed one has “almost” density of the pre-orbit of any point.

This thesis is divided in three chapters, each one with a brief introduction giving the

main goals of every chapter. We also deﬁne many of the concepts involved thro ughout the

work and pose many questions and remarks related to our results.

In the ﬁrst chapter, we address the main problem, to ﬁnd necessar y and suﬃcient

conditions in order to have robust transitive endomorphisms. In section § 1.1.1, we present

the main result of this work:

Main Theorem Let f ∈ E

) be a volume expanding map satisfying the following

properties:

1. There is an open s et U

in T

such that f |

is expanding and diam

ext

) < 1.

2. {f

−k

(x)}

k≥0

is dense for every x ∈ T

Introduction v

3. There exists 0 < δ

< diam

int

) and there exists an open neighborhood U

of U

such that for every arc γ in U

with diameter larger than δ

, there is a point y ∈ γ

such that f

(y) i s not in U

for any k ≥ 1.

4. For every z ∈ U

, there exists ¯z ∈ U

such that f(¯z) = z.

Then, for every g close enough to f, {g

−k

(x)}

k≥0

is dense for every x ∈ T

We recommend to the reader before entering into the proof of the Main Theorem, to

give a glance to section § 1.1.2 in order to gain some insight about the proof. We want

to highlight that this theorem as it is enunciated, it is not assumed the existence of any

tangent bundle splitting. In the case that there exists a partially hyperbolic splitting

we may get the same conclusion but with wea ker hypotheses. This is given in Theorem

2 in section § 1.2. One question that arises from this formulation is: if a map satisﬁes

the hypotheses of the Main Theorem, is it true that th i s map is isotopic to an expanding

endomorphism ? The Main Theorem can be recasted in terms of the geometrical properties,

see Main Theorem Revisited in section § 1.1.8.

In the second chapter, we show some geometrical and topological consequences from

the Main Theorem. Besides, we study the existence of Markov Partitions for maps of our

type, via semiconjugation with linear expanding endomorphisms, and how it allows us to

extract some informat ion about the transitivity of the ma p isotopic to a linear expanding

endomorphism. Also, we pose some related questions.

In the third and last chapter, we construct examples of robust transitive endomor-

phisms verifying the hypotheses of the Main Theorem, the Main Theorem Revisited and

Theorem 2.

vi Introduction

Chapter 1

Main Result

In this chapter, our goal is to give suﬃcient conditions to get robust transitivity for

n−dimensional torus endomorphisms.

§ 1.1 Volume expanding endomorphisms without invariant

splitting

An endomorphism of a diﬀerentiable manifold M is a diﬀerentiable function f : M → M

of class C

with r ≥ 1. Throug hout this work we will assume the endomorphism f to be

a local diﬀeomorphism. That means that given a ny point x, there exists a n open set V

containing x such that f from V to f(V ) is a diﬀeomorphism. For the main result we

do no t assume the existence of any invariant splitting for f. Let us denote by E

(M) the

space of C

−endomorphisms of M endowed with the usual C

topology.

Before entering into the Main Theorem, let us recall some deﬁnitions t hat are involved.

Deﬁnition 1.1 (Volume expanding map)

We say that a map f is vo lume expanding if there exists σ > 1 such tha t |det(Df)| > σ.

2 1. Main Result

Deﬁnition 1.2 (Invariant set)

We say that a set Λ ⊂ M is a forward invariant se t for f ∈ E

(M) if f(Λ) ⊂ Λ and it

is invariant for f if f (Λ) = Λ.

Let us intro duce some notation that we will use throughout this work: if L : V → W

is a linear isomorphism between normed vector spaces, we denote by m{L} the minimum

norm of L, i.e. m{L} = L

−1



−1

Deﬁnition 1.3 (Expanding map)

We say that a map f of class C

is expanding in U a subset of M if there exists λ > 1

such that min

x∈U

{m{D

f}} > λ. It is said that a compact invariant set Λ is an expanding

set for an endomorphism f if f |

is an expanding map.

Deﬁnition 1.4 (Locally maximal set)

Let Λ be an expanding set for f ∈ E

(M). If there is an open neighborhood V of Λ

such that Λ =



k≥0

−k

(V ) then Λ is said to be locally maximal (or isolated) set. V is

called the isolating block o f Λ.

Deﬁnition 1.5 (Full orbit)

A sequence {x

}

k∈Z

is called a full orbit for f if f(x

) = x

k+1

for every k ∈ Z.

Deﬁnition 1.6 (Topologically transitive)

Let Λ be an invariant set for an endomorphism f : M → M. It is said that Λ is

topologically tra nsitive if there exists a point x ∈ Λ such that its forward orbit {f

(x)}

k≥0

is dense in Λ. We say that f is topologically tra nsitive if {f

(x)}

k≥0

is dense in M for some

x ∈ M.

Lemma 1 Let f : M → M be a continuous map of a locally compact separable metric

space M into itself. The map f is topologicall y transitive if and only if for any two

§ 1.1 Volume expanding endomorphisms without invariant splitting 3

nonempty open sets U, V ⊂ M, there exists a positive integer N = N(U, V ) such that

(U) ∩ V is nonempty.

Proof. See for instance [KH95, pp.29].

Deﬁnition 1.7 (Topologically mixing)

A topological dynamical system f : M → M is called topolog i cally mixing if for a ny

two nonempty open sets U, V ⊂ M, there exists a positive integer N = N(U, V ) such that

for every k > N the intersection f

(U) ∩ V is nonempty.

Deﬁnition 1.8 (Locally eventually onto)

A map f ∈ E

(M) is called locally even tually onto if for any nonempty open set

U ⊂ M, there exists a positive integer N = N(U) such that f

(U) = M.

Remark 1.1 In general, it holds that if a map is locally eventually onto, then it is

topologically mixing. If a map is topologically mixing, then it is topologically transitive.

The reciprocals are not true for endomorphisms case.

Remark 1.2 For endomorphisms, if the pre-orbit of every point is dense in the ma-

nifold, then the map is transitive. Note that if there exists just one point whose pre-or bit

is dense, it is not enough to conclude that the map is transitive.

Deﬁnition 1.9 (Robustly transitive)

The set Λ

(U) =



n∈Z

(U) is C

−robustly transitive if Λ

(U) =



n∈Z

(U) is

transitive for every endomorphism g C

−close enough to f. It is said that a map f is

−robustly transitive if there exists a C

neighborhood U(f) such that every g ∈ U(f )

is transitive.

4 1. Main Result

Deﬁnition 1.10 (Robust non existence of splitting)

We say that f restricted to an invariant set Λ has no splitting in a C

−robust way if

there exists a C

open neighborhood U(f) o f f such that for every g ∈ U(f) the tangent

space T Λ does not admit invariant subbundles.

Theorem 1 Let f ∈ E

(M) be a local diﬀeomorphism and U open s et in M such that

(U) =



n∈Z

(U) is C

−robustly transitive set and it has no splitting in a C

−robust

way. Then f is volume expanding.

Proof. The proof of this theorem is similar to the one of Theorem 4 in [BDP03,

pp.361], nevertheless we include the main steps of the proof.

Let us consider f ∈ E

(M) a local diﬀeomorphism and denote by Λ

(U) the C

−robustly

transitive (nontrivial) set for f, note that U could be the ent ir e manifold. The idea

of the proof is to assume that f is no t volume expanding and show that for every

−neighborhood of f U(f) ⊂ E

(M), t here exists ψ ∈ U(f ) such that ψ has a sink

and therefore ψ cannot be tr ansitive.

Suppose that f is not volume expanding. Since f is onto, it cannot be uniform volume

contracting in the entire manifold, so there are points in the manifold such that we have

expansion, i.e. 1 ≤ |det(Df

(x))| for some k ≥ 0, but it does not expand too much, i.e.

|det(Df

(x))| ≤ 1 + ǫ, with ǫ small. Then there are sequences x

∈ Λ

(U), k

∈ N and

> 1, with k

→ ∞ and τ

→ 1

, such that

1 ≤ |det(Df

))| < τ

This is equivalent to say that

−1



i=0

log(| det(Df(f

)))|) < log(τ

§ 1.1 Volume expanding endomorphisms without invariant splitting 5

We may take k

such that f

) = f

) for all i = j, i, j ∈ {0, . . . , k

}. Consider for each

n t he Dirac measure δ

supported in { x

, f(x

), . . . , f

)}, i.e. δ



−1

i=0

)

As the space of probabilities is compact with the weak star to pology, there exists a subse-

quence of {δ

}

that converges to a n invariant probability measure µ such that



log |det(Df(x))|dµ(x) ≤ 0.

In fact, a classical ar gument proves tha t µ is invariant by f, since f

∗

(µ) − µ is the weak

star limit of

(δ

)

− δ

), which converge to zero. Observe that



log |det(Df(x))|dδ

−1



i=0

log(| det(Df

))|) =

log(| det(Df

))|) ≤ log(τ

then because τ

→ 1

we deduce that



log |det(Df(x))|dµ(x) ≤ 0.

By the ergodic decomposition theorem, there is an ergodic and f−invariant measure ν

such that



log |det(Df(x))|dν(x) ≤ 0.

Using the ergodic closing lemma for nonsingular endomorphisms, given ε > 0 there is

g close to f and a g−periodic point y such that

−1



i=0

log(| det(Dg(g

(y)))|) < ε,

where m

is the period of y. Note that if ε → 0, then m

→ ∞. So, taking ε > 0

arbitrarily small and m

big, using Franks’ Lemma [Fra71] we get ϕ close to g such that

(y) = y ∈ Λ

(U) and

−1



i=0

log(| det(Dϕ(ϕ

(y)))|) < 0,

6 1. Main Result

this means t hat |det(Dϕ

(y))| < λ < 1. Observe that we are assuming the dimension of

the manifold greater or equal to 2, so t he fact that the modulus of the jacobian of ϕ be

lower than 1 does not imply that all the eigenvalues have modulus smaller than 1.

Since Λ

(U) is C

−robustly transitive, after a perturbation, we may assume that the

relative ho moclinic class H(y, ϕ, U) of y is the whole Λ

(U). Now, consider the dense

subset Σ ⊂ Λ

(U) consisting of all the hyperbolic periodic points of Λ

(U) homoclinically

related to y.

If ϕ is close enough to f, then the tangent bundle does not admit a splitting a s well.

Using the idea of the proof of Lemma 6.1 in [BDP03, pp. 40 7] and, after that, Franks’

Lemma, we obtain that there exists ψ a perturbation of ϕ a nd a point p ∈ Σ such that all

the eigenvalues of Dψ

m(p)

(p) have modulus strictly lower than 1, where m(p) is the period

of p. This means that the maximal invariant set in U of ψ contains a sink, but this is a

contradiction since we choose ψ suﬃciently close to f such that Λ

(U) is still transit ive.

Remark 1.3 If Λ

(U) admits a splitting, then the extremal indecomposable subbun-

dle is volume expanding.

Remark 1.4 Theorem 1 implies t hat volume expanding is a necessary condition for

an endomorphism, which is local diﬀeomorphism, to be a robust transitive map. However,

volume expanding is not a suﬃcient condition that guarantees robust transitivity for a

local diﬀeomorphism. For instance, consider a product of an expanding endomorphism

times an irra tional rot ation: this map is volume expanding and transitive but not robust

transitive.

Remark 1.5 It is expected that if f is r obust ly t r ansitive and has no invariant sub-

bundles in a robust way, then f is a local diﬀeomorphism. It depends on whether the

Ergodic Closing Lemma holds even if there ar e critical points, since for maps with critical

§ 1.1 Volume expanding endomorphisms without invariant splitting 7

points already exists a version of Connecting Lemma, Closing Lemma and Franks’ Lemma,

which are the principal results involved in the proof o f Theorem 1.

Henceforth, we work in the n−torus T

. Since in dimension one volume expanding

endomorphism is equivalent to be expanding map, we also assume tha t the dimension n

is at least 2.

Deﬁnition 1.11 (Internal diameter)

Let U be an o pen set in T

. Denote by



U the lift of U restricted to a fundamental

domain. Deﬁne the internal diameter of U

diam

int

)= min

k∈Z

\{0}

dist(



U + k),

where dist(A, B) := inf{max

1≤i≤n

− y

| : x = (x

, . . . , x

) ∈ A, y = (y

, . . . , y

) ∈ B}.

Deﬁnition 1.12 (External diameter)

Let U be an open set in T

. Denote by



U the lift o f U restricted to a fundamental

domain. We say that externa l diameter of U is less than 1, denoting by diam

ext

(U) < 1,

if the closure of



U is contained in the interior of [0, 1 ]

Remark 1.6 Observe t hat volume expanding implies that the map is a local diﬀeo-

morphism.

§ 1.1.1 The Main Result

Our main result gives suﬃcient conditions for volume expanding endomorphisms to be

robustly transitive, independently of the existence or not of an invariant splitting.

8 1. Main Result

Main Theorem Let f ∈ E

) be a volume expanding map, n ≥ 2, satisfying the

followi ng properties:

1. There is an open s et U

in T

such that f |

is expanding and diam

ext

) < 1.

2. {f

−k

(x)}

k≥0

is dense for every x ∈ T

3. There exist 0 < δ

< diam

int

) and an open neigh borhood U

of U

such that for

every arc γ in U

with diameter larger than δ

, there i s a point y ∈ γ such that f

(y)

is not in U

for any k ≥ 1.

4. For every z ∈ U

there exists ¯z ∈ U

such that f(¯z) = z.

Then, f or every g close enough to f, { g

−k

(x)}

k≥0

is dense for every x ∈ T

Let us point out the following observations:

Remark 1.7 The hypothesis of external diameter less than 1 and hypothesis (4) are

technical. This means that they are necessary conditions for proving our result, but we

do not know if there exist weaker conditions that implies the thesis of o ur theorem.

Remark 1.8 The condition diam

ext

) < 1 implies that the closure of



is contained

in the interio r of [0, 1 ]

, where



is the lift of U

restricted to [0, 1]

. Note that U

not need to be simply connected and could have ﬁnitely many connected components.

Actually the important fact is that the closure of the convex hull of the lift of U

restricted

to [0, 1]

is still contained in (0, 1)

. Moreover, diam

int

) = diam

int

), where U

is the

convex hull of



Remark 1.9 If f ∈ E

) satisﬁes the hypotheses of the Main Theorem, then f is

topologically transitive. Indeed, as a consequence of the Main Theorem, f is C

robustly

topologically transitive.

§ 1.1 Volume expanding endomorphisms without invariant splitting 9

Remark 1.10 The Main Theorem is formulated fo r n−dimensional torus but we be-

lieve that it can be extended to any manifold tha t at least supports an expanding endo-

morphisms, for example nilmanifolds (see [Shu69]).

Remark 1.11 Λ



n≥0

−n

) is an expanding set. Moreover, we know that

given any arc γ in U

with diameter greater than δ

, there exists a point x ∈ γ such that

(x) is not in U

for any k ≥ 1. Therefore, γ ∩ Λ

= ∅ and Λ

is not trivial.

Remark 1.12 Using hypothesis (4) of the Main Theorem, given any point x ∈ U

, we

can construct a sequence {x

}

k≥0

such that x

= x, x

∈ U

and f(x

k+1

) = x

for every

k ≥ 0. We call this sequence by inverse path.

Remark 1.13 Let us denote Λ



n≥0

−n

). This set has t he following proper-

ties:

1. Λ

is an expanding set.

2. By hypothesis (3) of the Main Theorem, given any arc γ in U

with dia meter greater

than δ

, there exists a point x ∈ γ such that f(x) ∈ Λ

3. Since the hypothesis 0 < δ

< diam

int

) is an open condition, we may take U

open neighborhood of U

such that δ

< diam

int

) < diam

int

). Then for every

arc γ in U

with diameter greater than δ

holds that γ ∩ Λ

is non empty.

4. Λ

is invariant, i.e. f(Λ

) = Λ

. It is clear that Λ

is forward invariant. So let

us prove that Λ

⊂ f (Λ

). Pick a point x ∈ Λ

and consider the sequence {x

}

k≥0

given by remark (1.12). Let us show that x

∈ W for any k ≥ 0, where W =

∪

n≥0

−n

) = Λ

. If this is not true, there exist k ≥ 0 and n

≥ 0 such t ha t

) ∈ U

. First, observe that remark (1.12) implies that f

) = x

k−n

for 0 ≤

n ≤ k. In particular, f

) = x

if k ≥ 0. And f

) = f

n−k

)) = f

n−k

)

10 1. Main Result

for n > k ≥ 0 . Therefore, if −k ≤ −n

≤ 0, then f

) = x

k−n

. Since every x

belongs to U

, we obta in that f

) belongs to U

which is a contradiction because

it was supposed that f

) ∈ U

. If −n

< −k < 0, then f

) = f

−k

Since x

∈ Λ

, every positive iterat e of x

by f belongs to U

, thus f

) ∈ U

which contradicts the fa ct that f

) ∈ U

. Thus, x

∈ Λ

for every k ≥ 0.

5. In section § 1.1.3, we prove that this set is locally maximal or it is contained in an

expanding locally maximal set.

Question 1.1 If f satisﬁes the hypotheses of the Main Theorem, does this implies

that f is isotopic to an expanding endomorphism?

§ 1.1.2 Sketch of the Proof of Main Theorem

We want to prove that any small perturbation g of t he initial system f has the property

that the pre-orbit of any point is dense in the ma nif old. The mechanism to prove that

is the following: g i ven any open set V, there exist x ∈ V and k ∈ N such that g

(V )

contains a ball of a ﬁxed radius R

centered in g

(x). If we have the latter property, we

may conclude our claim, since given 0 < ε < R

for g ε/2−close to f the pre-orbit of any

point by g are ε−dense, hence g

(V ) intersects {g

−n

(z)} for any z. Therefore, V intersects

−n

(z)} for any z.

Note that because f is expanding outside U

, if x is a point in U

such that its forward

orbit stays outside U

, then f

(x)) ⊃ B

(x)), where λ

> 1 is the expanding

constant of f. The goal is to show that for any open set V , there exists a point x ∈ V

such that t he forward o r bit of some iterate f

(x) stay outside U

Now, working in the covering space, since U

has external diameter less than 1 and the

volume increases, follows tha t the diameter of the forward iterates of the lift of V grows.

Hence there is so me iterate with diameter big enough such that we may use hypothesis

§ 1.1 Volume expanding endomorphisms without invariant splitting 11

(3) to o btain such a point x. The a im is to show that this mechanism is r obust .

In order to prove the statement we use a geometrical approach:

1. Hypothesis (3 ) implies t ha t there is an expanding subset that “separates”, meaning

that a nice class of ar cs in U

intersects this set. (See Lemma 3 in section § 1.1.5)

2. Properly chosen, this set is locally maximal. (See Lemma 2 in section § 1.1.3)

3. Hence, it has a continuation conjugated to the initial one. (See Claim 1.1 in section

§ 1.1.4)

4. Therefore, that topolog ical property of separation persists. (See Lemma 4 in section

§ 1.1.5)

Finally, since the initial system has the pre-orbit of any point dense, the perturbed has

the pre-orbit o f any point “almost” dense. Then using the geometrical appro ach as above

we conclude the density of the pre- orbit of any point .

§ 1.1.3 Existence of a Locally Maximal Set for f

Lemma 2 Either Λ

is a locall y m aximal set or there exists Λ

∗

an expanding locally

maximal set for f such that Λ

⊂ Λ

∗

and Λ

∗

veriﬁes that every arc γ in U

with diameter

greater than δ

has a point such that the image by f belongs to Λ

∗

. Moreov e r, every arc γ

in U

with diameter greater than δ

intersects Λ

∗

Proof. We may divide the proof in two cases:

Case I . Λ

∩ ∂U

= ∅.

Let us observe that Λ

∩∂U

= ∅ implies that Λ

is contained in the open neighbor hood

V = int(U

). Then V is an isolating block for Λ

, therefore Λ

is locally maximal.

12 1. Main Result

Case II. Λ

∩ ∂U

= ∅.

Choose ε > 0 suﬃciently small such that the open ball B

(x) is cont ained in U

for all

x ∈ Λ

and for every x ∈ Λ

, since f is a local diﬀeomorphism, there exists an open set U

such that f |

: U

→ B

(x) is a diﬀeomorphism. No t e that the collection {B

(x)}

x∈Λ

an open cover of Λ

. Since Λ

is compact, there is a ﬁnite subcover, let us say {B

)}

i=1

Fix λ

−1

< λ

′

< 1, where λ

is the expansion constant of f and pick N

′

greater or

equal to N, the cardinal of the ﬁnite subcover of Λ

, such that for every y ∈ Λ

, there is

i = i(y) ∈ {1, . . . , N

′

} such that B

′

(y) ⋐ B

), i.e. B

′

(y) ⊂ B

Let us deﬁne W =

′



i=1

) a nd



W =

′



i=1

By remark (1.13) Λ

is invariant, then we have that for every x

, there exists at least

one x

∈ Λ

such that f(x

) = x

. Let us consider for every 1 ≤ i ≤ N

′

all the possible

pre-images by f of x

that belongs to Λ

, i.e. recall that f is a local diﬀeomorphism,

hence for every point x ∈ M, the cardinal ♯{f

−1

(x)} = N

is constant, then for every

i ∈ {1, . . . , N

′

}, there exist K

⊂ {1 , . . . , N

} such that if j ∈ K

then x

∈ Λ

and

f( x

) = x

. Therefore fo r every i ∈ {1, . . . , N

′

} and for every j ∈ K

, there exist open sets

such that x

∈ U

and f |

: U

→ B

) is a diﬀeomorphism. Given i ∈ {1, . . . , N

′

for every j ∈ K

consider the inverse branches, ϕ

i,j

: B

) → U

such that

i,j

) = x

fϕ

i,j

(x) = x, ∀x ∈ B

Now, consider Λ

∗



n≥0

−n

(



W ). Clearly, Λ

⊂ Λ

∗

⊂ U

and Λ

∗

is an expanding

set. In order to show that Λ

∗

is locally maximal, it is enough to show that Λ

∗

∩ ∂



W = ∅,

which is equivalent showing that f

−1

(



W ) is contained in W. Just to make more clear what

follows, let us rewrite f

−1

(



W ) in terms of the inverse branches,

−1

(



W ) = f

−1

(

′



i=1

)) =

′



i=1



j∈K

i,j

)).

§ 1.1 Volume expanding endomorphisms without invariant splitting 13

So, it is enough to show that ϕ

i,j

)) ⊂ B

i,j

), for some x

i,j

∈ {x

, . . . , x

′

In fact,

i,j

)) = U

⊂ B

−1

(ϕ

i,j

)) ⊂ B

′

(ϕ

i,j

)) = B

′

)

then, there exists m

i,j

∈ {1, . . . , N

′

} such that B

′

) ⋐ B

i,j

), and the assertion

holds. Easily follows that Λ

∗

has the property that every arc γ in U

with diameter la rger

than δ

has a point such that it s image by f belongs to it.

Figure 1.1: Λ

looks like a ne t which is an expanding set that “separates”

Remark 1.14 We want to highlig ht that for diﬀeomorphisms there exist examples

of hyperbolic sets that are not conta ined in any locally maximal hyperbolic set, see for

instance [Cro02] and [Fis06]. A similar construction seems feasible for endomorphisms.

The hypothesis (4) guarantees that Λ

is an invariant set. Moreover, we can co nsider a

ﬁnite covering {B

)}

′

i=1

for Λ

, with x

∈ Λ

, in such a way tha t for every point y ∈ Λ

there is x

such that B

′

(y) ⊂ B

). Thus we conclude that Λ

∗

is contained in the interior



W and therefore the expanding set Λ

is either locally maxima l or is contained in a

locally maximal expanding set.

14 1. Main Result

§ 1.1.4 Continuation of the Locally Maximal Set

Deﬁnition 1.13 (δ−pseudo orbit)

The sequence {x

}

n∈Z

is said to be a δ−pseudo orbit for f if d(f(x

), x

n+1

) ≤ δ for

every n ∈ Z.

Deﬁnition 1.14 (ε−shadowed)

We say that a δ−pseudo orbit {x

}

n∈Z

for f is ε−shadowed by a full orbit {y

}

n∈Z

for

f if d(y

, x

) ≤ ε for every n ∈ Z.

Deﬁnition 1.15 (Topological conjugacy)

f : M → M is topologically conjugate to g : N → N if there exists a homeomorphism

h : M → N such that h ◦ f = g ◦ h.

In o rder to ﬁx some notation for what follows, we will denote by Λ

the expanding

locally maximal set f or f, it means that Λ

is either Λ

, in the case it is locally maximal,

or it is Λ

∗

given in Lemma 2; and denote by U the isolating block of Λ

Claim 1.1 There exists V

(f) an open neighborhood of f in E

(M) such that if g ∈

(f), then g is expanding on Λ



n≥0

−n

(U) and there exists an homeomorphism

: Λ

→ Λ

that gives the topological conjugacy and h

is closed to the identity.

Proof. In order to get the conj ugacy we use the Shadowing Lemma for expanding

endomorphisms, see for instance [Liu91].

Since Λ

is an expanding locally maximal set for f, there exists β > 0 such that f is

expansive with constant β in Λ

Fix 0 < η < β. By the endomorphism version of the Shadowing Lemma, there exists

ε > 0 such that a ny ε−pseudo orbit for f within ε of Λ

is uniquely η−shadowed by a full

§ 1.1 Volume expanding endomorphisms without invariant splitting 15

orbit in Λ

Take N such that



j=0

−j

(U) ⊂ {q : d(q, Λ

) < ε/2}.

There exists a C

neighborhood V(f) of f such that fo r g in V(f)



j=0

−j

(U) ⊂ {q : d(q, Λ

) < ε/2}

and for any x ∈



j=0

−j

(U), we may consider {x

}

n∈Z

a full orbit for g, where x

= x,

getting that {x

}

is an ε−pseudo o rbit for f.

Let Λ



n≥0

−n

(U). Taking an open subset V

(f) of V(f) small enough in the C

topology, then f or g ∈ V

(f), Λ

is an expanding locally maximal set for g. If g is close

enough to f, then g is also expansive with constant β. Moreover, the Shadowing Lemma

also holds fo r g.

Take g ∈ V

(f). Given x ∈ Λ

, consider {x

}

n∈Z

a full orbit for g, where x

= x.

As {x

}

is an ε−pseudo orbit for f, there exists a unique full o r bit {y

}

n∈Z

for f with

= y ∈ Λ

that η−shadows {x

}

n∈Z

Let us deﬁne h

: Λ

→ Λ

by h

(x) = y, where y is given by the Shadowing Lemma.

By the uniqueness of the shadowing point, this map is well deﬁned. The continuity of h

follows from the shadowing lemma.

Moreover, h

◦ g = f ◦ h

. In fact, consider the sequence {z

}

n∈Z

where z

= g(x

) =

n+1

. This ε−pseudo-orbit is η−shadowed by a unique full orbit {w

}

n∈Z

for f, with

= w ∈ Λ

. Then, for every n ∈ Z,

d(w

, z

) = d(f

), x

n+1

) = d(f

)), g

(g(x

)))

= d(f

◦ g( x

)), g

(g(x

))) = d(f

n+1

◦ f

−1

◦ h

◦g(x

), g

n+1

)) < η

Observe that f

−1

◦ h

◦ g(x

) = w

−1

is η−shadowing x

. So, by uniqueness, we have

that f

−1

◦ h

◦ g( x

) = y

; i.e. h

◦ g( x) = f ◦ h

(x).

16 1. Main Result

Since we can apply the Shadowing Lemma for Λ

using the same constants as in the

construction of h

, we deﬁne a map h

: Λ

→ Λ

such that h

◦ f = g ◦ h

. In fact,

if {y

}

n∈Z

is a full or bit for f with y

∈ Λ

, then it is an ε−pseudo orbit for g. Hence,

this pseudo orbit is uniquely shadowed by a full orbit {x

}

n∈Z

for g, with x

∈ Λ

. Thus,

) = x

and d(y

, x

) < η f or every n ∈ Z; moreover, h

is continuous and satisﬁes

◦ f = g ◦ h

just as h

Next, let us verify t hat h

is one to one. Let p

, p

∈ Λ

be two points such that

) = h

). Note that d(f

)), g

)) < η and d(f

)), g

)) < η by

construction. Then h

) is η−shadowed by p

and p

, which by uniqueness gives that

= p

Finally, for y ∈ Λ

, consider a full orbit of h

(y) by g. Since d(g

(y)), f

(y)) is

small for all n and some f f ull orbit of y shadows the g full orbit of h

(y), we have that

(y)) = y. Hence, h

is onto and therefore is a homeomorphism.

The next claim is a version for expanding endomorphisms that was already provided

for the case of hyperbolic diﬀeomorphisms in [R ob76, Theorem 4.1]. The goal is to show

that we can extend the conjugation between f|

and g|

to an open neighborhood U of

in such a way that still is an homeomorphism that conjugate f|

and g|

, noting that

the conjugation is unique just in Λ

. We are g oing to use this extension in next section

for proving the robustness of the property of Λ

disconnects a “nice” class of sets.

Claim 1.2 The homeomorphism h

: Λ

→ Λ

in claim (1.1) can be extended as an

homeomorphism H to an open neighborhood of Λ

such that H ◦ f = g ◦ H.

Proof. This g eometrical proof is inspired in t he proof given by Palis in [Pal68] and

also used to prove the Grobman-Hartman Theorem in [Shu87, pp.96].

Other alternative proof consist in using inverse limit space, by this an expanding

§ 1.1 Volume expanding endomorphisms without invariant splitting 17

endomorphism becomes a hyperbolic diﬀeomorphism and so Theorem 4.1 in [Rob76] could

be applied.

The goal is to choose U an isolating neighborhood of Λ

and to construct an homeo-

morphism fr om U ont o itself, using the inverse branches of f and g, and a fundamental

domain D

for f, i.e. for every x ∈ U \ Λ

, there exists n ∈ N such that f

(x) ∈ D

Observe that the isolating block of Λ

is also an isolating block of Λ

. Now we can take

in the same way a fundamental domain for g, D

. After it is taken an homeomorphism H

between both fundamental domains D

and D

. Then this homeomorphism is saturated

to U \ Λ

by backward iteration, i.e. if x ∈ U \ Λ

, let n be such that f

(x) ∈ D

, take

H ◦ f

(x) and then g

−n

◦ H ◦ f

(x) where g

−n

is taken carefully using the corresponding

inverse branches.

Denote by N

the cardinal of {f

−1

(x)}, since f is a local diﬀeomorphism, N

is cons-

tant. Let K ⊂ {1, . . . , N

} be such that for every i ∈ K, there exist U

⊂ U and

: U → U

inverse branch of f such that ϕ

(U) = U

and f(U

) = f ◦ϕ

(U) = U. Also,

for g as in claim (1.1), for every i ∈ K, there exist U

⊂ U and ϕ

: U → U

the inverse

branch of g such that ϕ

(U) = U

and g(U

) = g ◦ ϕ

(U) = U.

We wish to construct a n homeomorphism H on U satisfying H ◦ f = g ◦ H and

H |

= h

. We can begin as follows. Suppose t hat the restriction H : ∂U → ∂U is any

well-deﬁned orientation preserving diﬀeomorphism. The restriction of H to ∂U

is then

deﬁned as follows H(x) = ϕ

◦H ◦f (x) if x ∈ ∂U

because H conj ugate f and g. Extend

H to a diﬀeomorphism which send U \∪

i∈K

bounded by ∂U and ∂U

onto U \∪

i∈K

bounded by ∂U and ∂U

. We may assume that the Hausdorﬀ distance between U and

is small, see Lemma 2, then the initial H is close to the identity. Let us say that

d(H(x), x) < η, where η > 0 is given arbitr arily.

Given i, j ∈ K, denote U

j,i

= ϕ

◦ ϕ

(U) and U

2 i

= U



j∈K

j,i

. If x ∈ ∂U

j,i

then

H(x) = ϕ

◦ ϕ

◦ H ◦ f

(x) ∈ ∂U

j,i

. If x ∈ U

2 i

\ Λ

then H(x) = ϕ

◦ H ◦ f (x) ∈ U

2 i

18 1. Main Result

Doing this process inductively we have that: Given i

, . . . , i

∈ K, denote U

,...,i

◦···◦ϕ

(U) and U

n (i

n−1

,...,i

)

= U

n−1

,...,i



∈K

,...,i

. If x ∈ ∂U

,...,i

then H(x) =

◦···◦ϕ

◦H ◦f

(x). If x ∈ U

n (i

n−1

,...,i

)

\Λ

then H(x) = ϕ

n−1

◦···◦ϕ

◦H ◦f

n−1

(x).

And H(x) = h

(x) if x ∈ Λ

Let us prove that H is continuous.

Given x ∈ Λ

, let (x

)

be a sequence in U \ Λ

such that x

→ x, when n → ∞. Let

us prove that H(x

) → H(x), when n → ∞.

First, consider {z

}

k∈Z

an f −full orbit in Λ

such that z

= x and for every n ∈ N,

consider {z

}

k∈Z

a full o rbit by f associated to each x

using t he corresponding inverse

branches (for the backward itera t es) given by the full orbit of x, where z

= x

. Since f

is continuous, for every k ∈ Z, we have that z

→ z

when n → ∞.

Note that for every n ∈ N, there exists k

> 0 such that z

∈ U \∪

i∈K

. Furthermore,

∈ U for every k ∈ [−k

, k

]. Since H ◦ f = g ◦ H, we get that H(x

) ∈



k=−k

(U).

Hence, for η and ε as in claim (1.1) and for every n ∈ N, we have that {z

}

k=−k

a ﬁnite ε−pseudo orbit for g and it is η−shadowed by a g−orbit of H(x

) until k

for

forward itera t es and −k

for backward iterates.

Observe that as m goes to inﬁnity, the ﬁnite pseudo orbit y

= {z

}

k=−m

becomes

longer. Consider now the sequence {y

}

. Then y

→ {z

}

k=−m

when n → ∞. Hence,

the sets of shadowing points of the ﬁnite pseudo orbits y

converge to the shadowing

point of the inﬁnite pseudo orbit {z

}

, then H(x

) → h

(x) = H(x) when n → ∞.

§ 1.1.5 The Locally Maximal Set “Separates”

The main goal of this section is to show that the locally maximal set for f has a topo-

logical property that persist under perturbation, roughly speaking means that Λ

and Λ

§ 1.1 Volume expanding endomorphisms without invariant splitting 19

disconnect small open sets. We prove that Λ

intersects “some nice” class of arcs in U

and which Λ

also intersects f or all g nearby f. The ﬁrst question that a r ise is: which

arcs belong to this “nice” class?, the second questions in the context of proving the Main

Theorem is: why is this property enough? and the third question is: why do the “nice

class” exist? All these questions are answered along the section, but to give some brief

insight about the main ideas observe that:

1. These “nice” arcs have the property that we can build a “nice cylinder” (see deﬁnition

1.18) containing the initial arc and Λ

“separates” (see deﬁnition 1.20) this cylinder

in a “robust way”.

2. It is enough t o consider these “nice” class of arcs to ﬁnish the proof of the Main

Theorem. Suppose we have the existence of this class of a r cs and suppose that given

any open set there is an iterate by g that has a “nice” arc. Then there is a point in

this iterate which its forward orbits stay in the expanding region, hence the internal

radius growth until a ﬁxed radius in ﬁnitely many it era t es. It allows us to conclude

the density of the pre-orbit of any point by the perturbation, just noting t hat the

density of every pre-orbit by the initial map implies ε−density of the pre-orbit by

the perturbed map.

3. We show in claim (1.3) that ever y large arc admits a “nice” arc.

Let us deﬁne the concepts involved in this section.

Deﬁnition 1.16 (Cylinder)

Given γ a diﬀerentiable arc and r > 0, it is said that C(γ, r) is a cylinde r centered at

γ with radius r if

C(γ, r) :=



x∈γ

([T

γ]

⊥

)

20 1. Main Result

where ([T

γ]

⊥

)

denotes the ball B

(x) centered at x with ra dius r intersected with [T

γ]

⊥

the orthogonal to the tangent to γ in x.

Deﬁnition 1.17 (Simply connected cylinder)

Given γ a diﬀerentiable arc and r > 0, it is said that a cylinder C(γ, r) is simply

conn ected if it is retractile to a point.

Remark 1.15 Fixed the radius, the cylinder could be not retractile to a po int. In

this case, working in the universal covering space, consider the convex hull of its lif t and

then project it on the manifold. We call the resulting set as simply con nected cylinder as

well and denote in the same way as above.

Deﬁnition 1.18 (Nice cylinder)

Given γ an arc and r > 0, it is said that a cylinder C(γ, r) is a nice cylinder if it

is simply connected cylinder and if x

and x

are the extremal points of γ then A :=

([T

γ]

⊥

)

⊂ ∂C(γ, r) and B := ([T

γ]

⊥

)

⊂ ∂C(γ, r). In this case, we say that A and B

are the top and bottom sides of the cylinder.

Remark 1.16 Note that in general a cylinder does not have top and bottom sides

and does not be simply connected.

Hereafter, ﬁx U

an open set such that U

⊂ U

and δ

< diam

int

) < diam

int

Let d

= d

, U

) > 0, where d

denotes the Hausdorﬀ metric, and let k ∈ N such that

′

= δ

< diam

int

Let us denote by



U the lift of U

, π the projection of R

onto M and U

the convex

hull of



U ∩ [0, 1]

. Consider P

) the projection of U

in the i−th coordinate in the

n−dimensional cube [0, 1]

. Since diam

ext

) < 1 and remark (1.8), for every 1 ≤ i ≤ n,

there exist 0 < k

−

< k

< 1 such that k

−

< P

) < k

. Note that 1 + k

−

− k

> δ

′

for

every i, because 1 + k

−

− k

> diam

int

) by construction.

§ 1.1 Volume expanding endomorphisms without invariant splitting 21

Let R

= {x ∈ R

: k

−

+ m < x

< k

+ m} with m ∈ Z, 1 ≤ i ≤ n and x

is the

i−th coordinate of x. Thus, U

⊂



m∈Z,1≤i≤n

. Denote by L

= {x ∈ R

: x

= k

} and

−

= {x ∈ R

: x

= k

−

}. Let

f be the lift of f.

The next claim answer the third question stated at the beginning of the section.

Claim 1.3 Let m > 2

√

n be ﬁxed. Given any arc γ in R

with diam(γ) > m, t here

exist an arc γ

′

⊂ γ, 1 ≤ i ≤ n and j ∈ Z such that ∂γ

′

∩ (L

+ j), ∂γ

′

∩ (L

−

+ j + 1)

and P

(γ

′

) ⊂ [k

+ j, k

−

+ j + 1]. Moreover, γ

′

admits a nice cylinder, γ

∗

= π(γ

′

) is in U

diameter of γ

∗

is larger than δ

and γ

∗

also admits a nice cylinder contained in U

Proof. Take γ an arc with diameter larger than m, then t he projection of γ in the

i−th coordinate contains an interval of t he kind formed by k

and 1 + k

−

for some i

(or formed by k

+ j and k

−

+ j + 1 for some j ∈ Z). If it is not true, γ would be in

a n−dimensional cube with sides smaller than k

− k

−

< 1 and this cube has diameter

smaller than

√

n, but t his cont r adict the fact that diam(γ) > m. Hence, we may pick an

arc γ

′

in γ such that ∂γ

′

∩ L

+ j, ∂γ

′

∩ L

−

+ j + 1 and P

(γ

′

) ⊂ [k

+ j, k

−

+ j + 1]

for some 1 ≤ i ≤ n and some j ∈ Z. Therefore, diameter of γ

′

is greater than δ

and in

consequence its projection in M also has diameter greater than δ

Moreover, since the property of the arc γ

′

to have a projection in between k

+ j and

−

+1 +j, we may construct a cylinder centered at γ

′

and radius

such that this cylinder

is “f ar” away fro m



U, so this cylinder could b e simply connected or, if it is not simply

connected cylinder, it has ho les that are diﬀerent from



U. In the case that the cylinder

is not simply connect ed, we consider the convex hull of the cylinder, since the original

cylinder is bounded by L

+ j a nd L

−

+ j + 1, then the convex hull stay in between these

two hyperplanes and therefore it does not intersect



U. By abuse of notation, let us denote

this set by C(γ

′

), it is a simply connected cylinder. Observe tha t by construction, this

cylinder will have top and bottom sides, thus C(γ

′

) is a nice cylinder.

22 1. Main Result

Take γ

∗

= π(γ

′

), note tha t γ

′

can be choose such that γ

∗

is cont ained in U

and the

diameter of γ

∗

is larger than δ

, then proj ecting the nice cylinder for γ

′

in M we obtain a

nice cylinder for γ

∗

which is denoted by C(γ

∗

). This nice cylinder ha s the property that

every arc that goes from bottom to top side has diameter at least δ

and all this process

can be made in such a way that the nice cylinder is in U

Figure 1.2: Nice cylinders

Deﬁnition 1.19 (Lateral border)

Given γ a diﬀerentiable arc and r > 0. The lateral bord e r S of the cylinder C(γ, r) is

∂C(γ, r) minus the t op and bottom sides of t he cylinder if they exist.

Deﬁnition 1.20 (Separated horizontally)

We say that a nice cylinder C(γ, r) is separated horizontally by a set Λ if there exists a

connected component of Λ such that intersects the nice cylinder across the latera l border

and C(γ, r) minus that connected component of Λ has at least two connected component.

§ 1.1 Volume expanding endomorphisms without invariant splitting 23

Now, we are going to prove that the locally maximal set for f, found in section § 1.1.3,

has the geometrical property o f separating horizont ally these nice cylinders of a certain

radius.

Lemma 3 Given any arc γ i n U

with diame ter greater than δ

that admits a nice

cylinder as in claim (1.3) holds that Λ

separates horizontally this nice cylinder.

Proof. Let us denote by T the nice cylinder associated to γ as in the statement and

let A and B denote the top and bottom sides of T respect ively. Let ε > 0 be arbitrarily

small.

Let T

′

be a bigger cylinder containing T joint together with two security regions, denote

by S

and S

, and such that the distance between the lateral border of T and the lateral

border of T

′

is small, for instance d

(T, T

′

) =

, see ﬁgure (1.3). For security regions S

and S

we means two strips of

thickness glued to the sides A and B of T , or in other

words, S

(respectively S

) is the set of points in T

such that the distance from these

points to A (respectively B) is less or equal to

. This set T

′

was constructed in such a

way that its diameter is greater than δ

′

Since γ is in U

and diameter is greater than δ

, we can assure that T ∩ Λ

is non

empty. Consider all the connected components of T ∩Λ

. For every x ∈ T ∩Λ

, we assign

the connected component of T ∩ Λ

that contains x. Observe that we may deﬁne an

equivalence relatio n: x ∼ x

′

if and only if K

= K

′

. Then we pick one component from

each class, or in other words we pick just the connected components that are two by two

disjoints.

We claim that Λ

separates T hor izontally. If there exists one component K

that

separates T horizontally in more than one connected component , the assertion holds.

Suppose it does not happen, i.e. none of the K

separates T horizontally. Take U

open set in T

′

such that K

⊂ U

, ∂U

∩ Λ

= ∅, ∂U

is connected and ∂U

does not

24 1. Main Result

divides T horizontally. If there are many K

accumulating in one K

, then we could have

a same open set U

containing more than one connected component K

Observe that the collection {U

} is an open cover of T ∩Λ

. Since it is compact, there

is a ﬁnite sub cover {U

}

i=1

, i.e. T ∩ Λ

⊂ U = ∪

i=1

If the connected components of U does not separates horizontally T, it is easy to cons-

truct a curve going from A to B with diameter greater than δ

and empty intersection

with the U

’s; hence, this curve does not intersects the set Λ

. But this contradicts the

fact that every curve in U

with dia meter larger than δ

intersects Λ

. Then the connected

components of U separate T horizontally, denote by C

the connected components of T

minus these connected components of U that separates T horizontally.

Observe that every C

is path connected, since they are the complement of a ﬁnite

union of open sets in a simply connected set T . There exist a ﬁnite quantity of C

, let us

say m. We can reorder these sets enumerating from the top side. If we denote by V

each

of the connected compo nents of T ∩ U that separate T horizontally, we have two cases,

either C

is in between two consecutive V

and V

j+1

(or V

j−1

and V

) o r C

just intersects

one V

on the border.

The idea is to build a curve from top to bottom of T connecting C

with C

j+1

in such

a way that the diameter of the arc is greater than δ

but without intersecting Λ

, which

is an absurd because it is again in U

and has diameter greater t ha n δ

, then this curve

must intersects Λ

It is enough to show that we can pass from C

to C

j+1

without touching Λ

. For this,

we must observe tha t every V

is a union of ﬁnitely many U

, let us say U

, . . . , U

. Pick a

curve γ

in C

going from top to bottom, i.e. γ

goes from ∂V

to ∂V

j+1

(or ∂V

j−1

and ∂V

)

and γ

does not intersects the interior of V

and V

j+1

(or V

j−1

and V

), then there exists

∈ {i

, . . . , i

} such that γ

∩∂U

= ∅. After that continue this arc picking a curve follow-

ing by the border o f U

until C

j+1

, which has empty intersection with Λ

by construction,

§ 1.1 Volume expanding endomorphisms without invariant splitting 25

if it is not possible to do in one step, pick a nother U

and repeat the process. Note t ha t

this process ﬁnish in ﬁnitely many times. The resulting arc from j oint together all this

segment has diameter greater than δ

and with empty intersection with Λ

as we wanted.

Figure 1.3: Λ

splits ”horizontally” every nice cylinder in at least two connected com ponen t

Remark 1.17 In claim (1.1), remembering that d(h, id) < η, we may ﬁx η < min{

, δ

, β}.

So for this η, there exists ε

> 0 given by the shadowing lemma and this determine V

(f)

given in claim (1.1).

Lemma 4 Given g ∈ V

(f) and given γ an arc in U

with diameter greater than δ

′

such that it admits a nice cylinder C(γ,

), then γ ∩ Λ

is not empty.

Proof. Let g ∈ V

(f), it means that g is an endomorphism within distance of f less

than ε

in the C

topology. Take γ an arc in U

with diameter greater than δ

′

such that

C(γ,

) is a nice cylinder.

By construction, we may assume that every a r c taken in the nice cylinder that goes

from top to bottom has diameter greater or equal to the diameter of γ. Take two security

regions inside the cylinder, in the to p and bottom sides of the cylinder respectively, with

26 1. Main Result

of thickness each one, i.e. two strips glued to the top and bottom sides of the cylinder

such that each one is the set of points in the cylinder within distance to top (respectively

bottom) side less or equal to

, see ﬁgure (1.4). Let us denote by C

′

the cylinder resulting

of taking out these two security strips from the original cylinder C( γ,

), then t he diameter

of C

′

is still greater than δ

Hence, the dia meter of γ

′

= γ ∩C

′

is greater than δ

and it is in U

. Lemma 3 implies

that Λ

separates horizontally C

′

, hence γ

′

intersects Λ

, let us denote by x

the point in

the intersection.

Since x

∈ Λ

, claim (1.1) and remark (1.17), there exists x

∈ Λ

∩B

). Note that

separates B

) in a t least two connected component. Hence, Λ

separates B

) in

at least two connected component as well, because f |

and g |

are conjugated. There-

fore, Λ

must intersects γ.

Figure 1.4: Λ

intersects γ

§ 1.1.6 Getting Sets of Large Diameter

Lemma 5 There exist V

(f) and R > 0 such that for every g ∈ V

(f), if there is

x ∈ M such that g

(x) ∈ U

for every n ≥ 0, then there is ε

> 0 such that for eve ry

0 < ε < ε

, there exists N = N(ε) ∈ N such that B

(x)) ⊂ g

(x)).

§ 1.1 Volume expanding endomorphisms without invariant splitting 27

Proof. We may pick U

an open subset contained in U

such that m{Df |

} > λ

′

with 1 < λ

′

< λ

. Take V

(f) an open subset perhaps smaller than V

(f) such that

m{Dg |

} > λ

′

holds for every g ∈ V

(f). Let us ﬁx R = d

, U

) > 0.

Given 0 < ε < R, take N ∈ N such that (λ

′

)

−N

R < ε/2. Then B

(λ

′

)

−N

(x) ⊂ B

(x).

Observe that B

(x)) ∩ U

= ∅, for every n ≥ 0. Also,

(λ

′

)

−N

(x)) = B

(λ

′

)

−N+k

(x)) ⊂ B

(x)),

for every 0 ≤ k ≤ N. In particular, g

(λ

′

)

−N

(x)) ∩ U

= ∅, for every 0 ≤ k ≤ N. Then

(λ

′

)

−N

(x)) = B

(x)) ⊂ g

(x)).

Remark 1.18 Let us note that Lemma 5 holds for every point in Λ

Lemma 6 For every g ∈ V

(f) and given V an open path connected set in M, there

exists m

= m

(V, g) ∈ N such that diam(˜g

(



V )) > m, where ˜g and



V are the lift of g

and V, respectively. In particular, it contains an arc with diameter greater than m.

Proof. Let g ∈ V

(f) and V be an open path connected set in M. Since g is vo lume

expanding, let us say with expanding constant λ > 1, we have that vol(˜g

(



V )) > λ

vol(



V ),

for k ≥ 1. Iterating by ˜g, the volume increase and furthermore the diameter of its iterates

growth also in the cover ing space. Hence, there exists m

∈ N such that diam(˜g

(



V )) >

Remark 1.19 For the case that V is an open connected set, observe that given a point

in V there exists an open ball centered in this point and conta ined in V such that it is

28 1. Main Result

path connected. Then we may apply Lemma 6 to this ball and obta in a similar statement

for V .

§ 1.1.7 Proof of The Main Theorem

Let f ∈ E

) be such as in the statement . Lemma 2 implies that we may assume the

existence of Λ

an expanding locally maximal set for f.

Fix 0 < α < R, arbitra r ily small. Given x ∈ T

, since



i∈N

−i

(x)} is dense, there

exists n

∈ N such that



i=0

−i

(x)} is α/2-dense.

Take a neighborhood U(f) ⊂ V

(f), where V

(f) was given in Lemma 5, such that for

every g ∈ U(f) follows that



i=0

−i

(x)} is α/2-close to



i=0

−i

(x)}.

Hence,



i=0

−i

(x)} is α-dense.

Let V be an open connected set in T

. By Lemma 6, there exists m

∈ N such that

diam(˜g

(



V )) > m. Then we may pick an arc γ in ˜g

(



V ) with diameter larger than

m and applying claim (1.3) fo llows that there exists a connected piece γ

′

of γ such that

∗

= π(γ

′

) is in U

, diameter of γ

∗

is larger than δ

′

and it admits a nice cylinder C(γ

∗

By Lemma 4 follows that γ

∗

∩ Λ

is not empty, let y be a point in the intersection.

Hence, for this point y, there exists ε

= ε

(y) > 0 such that B

(y) ⊂ g

(V ), by

Lemma 5 taking 0 < ε < ε

, we get that there exists N = N(ε) ∈ N such that

(y)) ⊂ g

(V ).

§ 1.1 Volume expanding endomorphisms without invariant splitting 29

Hence, B

(y)) ⊂ g

(V ). Since the α−density, we have that



i=0

−i

(x)} ∩ B

(y)) = ∅.

Therefore, denoting by p = m

+ N,



i=0

−i

(x)} ∩ g

(V ) = ∅.

Taking the p−th pre-image by g, we obtain that there is i

∈ N such that



i=0

−i

(x)} ∩ V = ∅.

Thus, for every g ∈ U(f ) follows that



i∈N

−i

(x)} is dense in T

for every x ∈ T

(y)

(y))

∗

Figure 1.5: Iterations by the perturbed map

§ 1.1.8 The Main Th eorem Revisited

In this section, we enunciate a weaker ver sion of the Main Theorem. Observe that using

hypothesis (3) and (4) of the Main Theorem, we showed in sections § 1.1.3 and § 1.1 .5 the

30 1. Main Result

existence of a locally maximal expanding set for f which separates large nice cylinders,

and by section § 1.1.4 follows that this geometrical property persist under perturbation,

i.e. there is a set Λ

locally maximal which intersects a nice class of arcs in U

and

this property also holds fo r the perturbed. The hypothesis of f being volume expanding

guarant ees that given any open set in the covering space, we are able to choose some

iterates such that it contains an arc with diamet er big enough to apply claim (1.3) and

Lemma 3. Hence, the Main Theorem may be enunciated in a weaker version as fo llows:

Main Theorem Revisited Let f ∈ E

) be volume expandi ng such that the p re-

orbit of every point are dense. Suppose that there exist an open set U

with diam

ext

) < 1

and Λ

a locally maximal expanding set for f in U

such that every arc γ in U

with

diameter large enough intersect Λ

. Then, the pre-orbit of every point are C

robustly

dense.

Remark 1.20 Observe that the Main Theorem implies the Main Theorem Revisited,

the reciprocal could be false. The hypotheses of the Main Theorem assures the existence

of a nice Λ

with certain geometrical properties, but there might exist weaker conditions

that guarantee the existence of such a set.

§ 1.2 Volume expanding endomorphisms with invariant

splitting

Deﬁnition 1.21 (Unstable cone family)

Given f : M → M a local diﬀeomorphism, let V be an open subset of M such that

is a diﬀ eomor phism ont o its image. Denote by ϕ the inverse branches of f restricted

to V ; more precisely, ϕ : f(V ) → V such that f ◦ϕ(x) = x if x ∈ f(V ). A continuous cone

§ 1.2 Volume expanding endomorphisms with invariant splitting 31

ﬁeld C

= {C

}

deﬁned on V is called unstable if it is forwa r d invariant:

Df(x

′

) C

′

⊂ C

f(x

′

)

for every x

′

∈ V ∩ ϕ(V ).

Remark 1.21 Given a point x, there is not necessarily a unique unstable subbundle,

i.e. fo r each inverse path {x

}

k≥0

, it means x

= x and f(x

k+1

) = x

for k ≥ 0, there

exists an unstable direction belonging to C

Deﬁnition 1.22 (Complementary splitting )

We say that a splitting E

+ C

is comple mentary if the unstable cone C

contains an

invariant subspace whose dimension is equal to t he dimension of the manifold minus the

dimension of the central subbundle.

Deﬁnition 1.23 (Partially hyperbolic endomorphism with expanding

extremal direction)

It is said that an endomorphism f is partially h yperbolic with expandi ng extremal di-

rection provided the tang ent bundle splits into two non-trivial subbundles T M = E

⊕E

which are invariant under the t angent map Df, i.e. for every x ∈ M, there exists a com-

plementary splitting E

+ C

, where {C

}

is a family of unstable cones, and there exists

0 < λ < 1 such tha t for every inver se branches ϕ of f follows that

1. Dϕ(x) v < λ, for all v ∈ C

2. Df(x

′

) |

′

)

Dϕ(x)v < λ, for all v ∈ C

, where ϕ(x) = x

′

, f ( x

′

) = x.

§ 1.2.1 Theorem 2: Splitting Version

In section § 1 .1.1, we gives suﬃcient conditions for volume expanding local diﬀeomorphisms

without invariant subbundles be robustly transitive. Now, we state a version for the

32 1. Main Result

case when tangent bundle splits into two non-trivial subbundles, one with an expanding

behavior and the other one with nonunifo r m behavior but dominat ed by the expanding

one.

Theorem 2 Let f ∈ E

) be a locally diﬀeomorphism partially hyperbolic with ex-

pand i ng extremal direction satisfying the following properties:

1. {f

−k

(x)}

k≥0

is dense for every x ∈ T

2. There exist δ

> 0, λ

> 1 and k

∈ N such that fo r every x ∈ T

, if γ is a dis c

tangent to the unstable co ne C

with internal diame ter larger than δ

, there exists a

point y ∈ γ such that m{Df

(y))

} > λ

, for all i > 0, for all k > k

Then, f or every g close enough to f, { g

−k

(x)}

k≥0

is dense for every x ∈ T

§ 1.2.2 Proof of Theorem 2

The proof of Theorem 2 is pretty similar to the proof given in [PS06b], where it is proved

that any partially hyperbolic diﬀeomorphism satisfying a hypothesis like the one stated

in Theorem 2 and such that the strong stable fo liation is minimal, then the strong stable

foliation is r obust ly minimal. That property says that in any compact piece of the unstable

foliation, there exists a point such that the central bundle has uniform expanding behavior

along the forward o r bit , and this is exactly what we have. The key is to prove that this

property is robust under perturbation.

Given a local diﬀeomorphism f as in the statement of Theorem 2, we want to show

that any small perturbation g preserve the property of density of the pre-orbit of any

point. Our strategy is to prove that given any disc tangent to the unstable cones for g

with large enough internal diameter has a point such that the central direction along the

forward o rbit by g is uniformly expanding.

§ 1.3 Remarks About the Main Theorem and Theorem 2 33

Observe that given any open set, since we have a direction that is indeed expanding,

the diameter along the unstable direction o f the iterates growth. Then we are able to pick

a disc inside this iterate such that the disc is t angent to the unstable cones with diameter

large enough to apply the last property. Hence, there exists a point which its forward

orbit is expanding in all direction, then there is some iterat e such that it contains a ball

of a ﬁx radius ε.

Since g is close enough to f, we have that the pre-orbit by g are ε−dense. Therefore,

given any open set, by the property of the unstable discs, there exists an iterate such that

it intersects the pre-orbit by g of any point. Thus, we conclude the density of the pre-orbit

of any point by the perturbation.

Moreover, the proof of Theorem 2 can also be performed in the spirit of Main Theorem.

In fact, it is possible to show that



l≥0

−l

({x : m{Df

(x))

} > λ

, n > 0, k > k

})

is an invariant expanding set such that separates unstable discs. This provides a geome-

trical interpretation.

§ 1.3 Remarks About the Main Theorem and Theorem 2

On regard the similarities between Main Theorem and Theorem 2, we must note that in

both theorems we assume that f is volume expanding, since we know by Theorem 1 that

volume expanding is a necessary condition in order to have robust transitivity. Also, we

assume that the pre-orbit by f of every point is dense, actually this hypothesis is stronger

than transitivity. Moreover, this hypothesis does not depend on the existence of splitting.

Besides, in the Main Theorem we asked for large arcs to contain points such that its

forward iterations remain in the expanding region. The same is required in Theorem 2 but

34 1. Main Result

just for unstable discs, and the equivalent for the splitting ver sion to say that the forward

orbit is “in an expanding region” is that the central bundle along the forward orbit of such

points has uniform expanding behavior.

The main diﬀerence in their proof arise from the fact that in the version with splitting,

since we know that we have uniform expansion in one direction, any disc with internal

diameter larger than δ

and tangent to this direction, growth until length δ

> δ

in a

bounded unifor m time, independently of the disc. Note that we cannot guarantee that

this happens just having volume expansion.

Observe that in the Main Theorem is not assumed that f does not have any splitting.

In fact, it could also be partially hyperbolic. However, knowing in adva nce that the

endomorphism is partially hyperbolic then it is possible to get suﬃcient conditions f or

robust transitivity weaker than the one requires by the Main Theorem.

Chapter 2

Exis tence of a semiconjugation to a

linear expanding endomorphism

In this chapter, we show some consequences fr om the Main Theorem. The main goals are

to study the properties of the semiconjugation between a linear expanding endomorphism

and an isotopic endomorphism to the initial one, and the relation between the Markov

Partition and Transitivity of these maps.

§ 2.1 Dynamical Consequences: Geometrical and Topological

Deﬁnition 2.1 (Totally disconnected set)

A set Λ is said to be totally disconnected if every point is a connected component.

By Lemma 3 we have that

Corollary 1 Λ

is not totally disconnected. Furthermore, Λ

intersects eve ry large

arc in U

. If the connected components of U

are simply connected, then Λ

is a basic set.

Remark 2.1 Since hypothesis (3) o f the Main Theorem, the connected components

of ∪

n≥1

−n

) have diameter less than δ

36 2. Existence of a semiconjugation to a linear expanding endomorphism

Remark 2.2 The fact of Λ

be expanding locally maximal set implies that there exists

a Markov Partition for this set.

§ 2.2 The Case f is Isotopic to a Linear Expanding

Endomorphism E

In this section, given an endomorphism f iso topic to a linear expanding endomorphism E,

we construct the semiconjugation between these two maps and study the properties of the

semiconjugation and the relation with the existence of Markov Pa r tition for f and how

it can help us to deduce if f is tr ansitive. Moreover, we are interested in the case t hat

we know that there exists an expanding locally maximal set Λ

with the properties given

in the Main Theorem and how this gives some informatio n about the Markov partition.

Finally, we pose some related questions.

§ 2.2.1 Existence of the Semiconjugation

Let E : T

→ T

be a linear expanding endomorphism and let f : T

→ T

be an

endomorphism isotopic to E.

Let us remember that the lift of f is given by

f =



E + p, where



E ∈ SL(n, Z) (called

the linear part) is the lift of E and p is Z

−periodic isotopic to a constant.

Lemma 7 There exists a semiconjugation between

f and



E; i.e. there exists H : R

→

continuous, onto and H ◦

f =



E ◦ H. Moreover, there exists a constant K

> 0 such

that H − I < K

. Also, for any m ∈ Z

we have that H(x + m) = H(x) + m.

Proof.

Since

f =



E + p and p is bounded, ﬁxing K > p, every full orbit { x

}

k∈Z

for

f is a

K−pseudo-orbit for



E. In fact, given x in R

, we may associate a full orbit {x

}

k∈Z

for

§ 2.2 The Case f is Isotopic to a Linear Expanding Endomorphism E 37

this means that

f(x

) = x

k+1

and x

= x, then for every k ∈ Z follows that





E(x

) − x

k+1

 = 



)) −

k+1

) = (



E −

f) (

))

≤ 



E −

f = p.

Note that



E is an expanding map and linear, therefore it has global product structure

and in consequence the (endomorphism version) shadowing lemma holds f or any pseudo

orbit of



E; that is, for any K > 0, there is K

= K

(K) > 0 such that for any pseudo

orbit, 



E( x

) − x

k+1

 < K for every k ∈ Z, there exists a unique y ∈ R

such that y

−shadows { x

}

k∈Z

. Then for every x there is a unique point y = H(x) such that





(H(x)) −

(x) < K

, ∀ k ∈ Z. (2.1)

As a consequence of the shadowing lemma, the map H : R

→ R

is well deﬁned and

continuous. Furthermore, H ◦

f =



E ◦ H and H − I < K

. Hence H is onto.

Now, given m ∈ Z

we have that





(H(x) + m) −

(x + m) = 



(H(x)) +



(m) −[



(x + m) + p

(x + m)]

= 



(H(x)) − [



(x) + p

(x)]

= 



(H(x)) −

(x) < K

, ∀ k ∈ Z.

Then by uniqueness follows that H(x + m) = H(x) + m.

Consider π : R

→ T

the canonical projection. Let us deﬁne h : T

→ T

h(π(x)) = π(H(x)). Hence, we have the f ollowing

Corollary 2 There exists a semiconj uga tion between f and E, i.e. there exists a map

h : T

→ T

continuous, onto and h ◦ f = E ◦ h.

Remark 2.3 The semiconjugation h has the following properties:

38 2. Existence of a semiconjugation to a linear expanding endomorphism

1. f transitive implies int(h

−1

(x)) = ∅ for every x ∈ T

. In fact, let us suppose

that there exists a point x such that the interior of h

−1

(x) is not empty, call

U =int(h

−1

(x)), so h(U)=x. Since f is t r ansitive, there is n > 0 such that f

(U)∩U

is nonempty. Then h(f

(U)) = E

(h(U)) = E

(x), hence h(f

(U)∩U) = x = E

(x).

Therefore, x is periodic and U is an open periodic set for f, but this is not possible

since f is tra nsitive.

2. h

−1

(x) is connected. Let π(˜x) = x, by the construction of h, h

−1

(x) is connected if

and only if H

−1

(˜x) is connected. So, let us suppose that H

−1

(˜x) is not connected,

then there exist ˜z, ˜y ∈ H

−1

(˜x) in two diﬀerent connected components, i.e. there

exists L hyperplane dividing R

in two component such that ˜z and ˜y are in diﬀerent

components. Since H(˜z) = H(˜y) = ˜x, we have that



(H(˜z)) =



(H(y)) =



(x),

for every n ≥ 0, then H(

(˜z)) = H(



(˜y)). Hence, by (2.1), 

(˜z) −

(˜y) ≤



(˜z)−H(

(˜z))+ H(



(˜y))−

(˜y) < 2K

. On the other hand, we have that for

any ˜w ∈ L, H( ˜w) = ˜x, then H(

( ˜w)) − H(

(˜z) ) = 



(H( ˜w)) −



(H(˜z)) ≥

H( ˜w) −H(˜z), with λ > 1. Since 

( ˜w) −

(˜z)  ≥  H(

( ˜w)) −H(

(˜z) )−

H(



( ˜w)) −

( ˜w) − H(



(˜z)) −

(˜z)  ≥ λ

H( ˜w) − H(˜z) − 2K

, we have

that the distance between

(˜z) and L goes to inﬁnity and the same happen for

(˜y). So, the distance between

(˜z) and

(˜y) cannot be bounded contradicting

that 

(˜z) −

(˜y) < 2K

. Thus, H

−1

(˜x) is connected and therefore h

−1

(x) is

connected.

§ 2.2.2 Markov Partition and Transitivity

Let us deﬁne Markov Partition for an endomorphism

Deﬁnition 2.2 (Markov partition for endomorphism)

A Markov Partition for an endomorphism E : M → M is a family P = {R

, . . . , R

}

§ 2.2 The Case f is Isotopic to a Linear Expanding Endomorphism E 39

of compact sets covering M such that:

1. For every 1 ≤ i < j ≤ N, R

∩ R

= ∂R

∩ ∂R

2. For all 1 ≤ i ≤ N, E |

int(R

)

is injective.

3. For all 1 ≤ i ≤ N, int(R

) = R

4. If E(R

) ∩ int(R

) = ∅, then R

⊂ E(R

Remark 2.4 All expanding endomorphism has a Markov Partition.

Let P

be a Markov Partition of the linear map E, denote by R

every element of the

partition. Then T



i=1

. Consider the fo llowing family

:= {h

−1

) : i = 1, . . . , N}.

Claim 2.1 P

is a Markov Partition for f.

Proof. Let us denote by

= h

−1

• First, observe that int(R

)∩int(R

) = ∅ if i = j, so int(

)∩int(

) = h

−1

(int(R

))∩

−1

(int(R

)) = ∅. Then

∩

= ∂

∩ ∂

• f |

int(

)

is a diﬀeomorphism. Let us suppose that f |

int(

)

is not an homeomorphism.

So, take x, y ∈ int(

) such that f(x) = f (y). Note that h(x) and h(y) belong to int(R

where E restricted to it is injective, then E ◦ h(x) = h ◦ f (x) = h ◦ f(y) = E ◦ h(y)

imply that h(x) = h(y). Pick a curve γ in int(

) from x to y, then f(γ) is a curve with

homology, and therefore h(f( γ)) has homology also. On the other hand, h(γ) does not

have homology and E |

int(R

)

is an homeomorphism, then E(h(γ)) = h(f (γ)) does not have

homology as well.

• int(

) =

holds for all 1 ≤ i ≤ N.

40 2. Existence of a semiconjugation to a linear expanding endomorphism

• If f(

) ∩int(

) = ∅, then

⊂ f(

). Suppose that y ∈ f(

) ∩int(

), therefore

h(y) = h ◦f (¯y) = E ◦h(¯y) belongs to E(R

) ∩int(R

), where ¯y ∈

is such that f(¯y) = y.

Then holds that R

⊂ E(R

). Given ¯z ∈

, there exist z ∈ R

and ˆz ∈ R

such that

¯z ∈ h

−1

(z) and E(ˆz) = z. Then ¯z ∈ h

−1

◦ E(ˆz) = f ◦ h

−1

(ˆz) , i.e. ¯z ∈ f (

For what follows, we study the r eﬁnement o f the partition of f induced by t he semicon-

jugation and see what information can be deduced from it , such as under which condition

on the reﬁnement can be concluded that h is an homeomorphism. Moreover, we give

necessary and suﬃcient conditions on the reﬁnement that characterize the transitivity of

Given a point x ∈ T

, denote by P

(x) t he element of the partition P

that contains

x. Let us consider a reﬁnement of the partition,

(x) = {y : f

(y) ∈ P

(x)), k = 0, . . . , n}.

Claim 2.2 P

(x) = h

−1

(h(x))) for every n ∈ N.

Proof. First observe that P

(h(x)) = {z : E

(z) ∈ R

(h(x))), k = 0, . . . , n},

where R

(w) denote the element of the partition for E that contains w.

Let us show ﬁrst that h

−1

(h(x))) ⊂ P

(x). Given z ∈ P

(h(x)), we have that

◦ h

−1

(z) = h

−1

◦ E

(z) ∈ h

−1

(h(x)))). Since E

(h(x)) = h(f

(x)), f

−1

(z)) ∈

(x)).

Now, let us prove the equality. Given y ∈ P

(x) follows that E

(h(y)) = h(f

(y)) ∈

h(P

(x))). Therefo r e, E

(h(y)) ∈ R

(h(f

(x))) = R

(h(x))). Thus, the equality

holds.

§ 2.2 The Case f is Isotopic to a Linear Expanding Endomorphism E 41

Let us deﬁne

∞

(x) :=



n≥0

(x).

Claim 2.3 P

(x) is a compact connected set for every n. Moreover, P

∞

(x) is a com-

pact connected set.

Proof. F ir st observe that

(x) = P

(x) ∩f

−1

(f(x))) ∩ . . . ∩ f

−n

(x)))

and f

−1

(

) ∩

is a compact connected set contained in

for every 1 ≤ i, j ≤ N. Then,

we get that P

(x) is a compact connected set. Let us assume that P

n−1

(x) is compact and

connected, then P

(x) = P

n−1

(x) ∩ f

−n

(x))) is a compact connected set, because

−n

(x))) has one connected component in P

n−1

(x) and it is compact as well.

Now, since {P

(x)}

is a nested sequence of compact and connected sets, we have that

∞

(x) is a compact connected set.

Claim 2.4 h is a conjugacy if and only if for every x ∈ T

, P

∞

(x) = {x}.

Proof. Take z ∈ P

∞

(x) and suppose that z and x are not equals. Since h is a

conjugacy, h(x) = h(z). Because E is a n expanding endomorphism, let us call λ

> 1 the

expanding constant, d(E

(h(x)), E

(h(z))) > λ

d(h(x), h(z)). Then, there exists k

> 0

such that E

(h(z)) ∈ R

(h(x))) for all k ≥ k

. On the other hand, z ∈ P

∞

(x) implies

that h(z) ∈ P

(h(x)), consequently E

(h(z)) ∈ R

(h(x))) for all k ≥ 0. Thus, x = z.

Now let us assume that P

∞

(x) = {x}. We have to prove that h is one-t o-one. Consider x

and y such that h(x) = h(y). Then h(y) ∈ P

∞

(h(x)). Thus, y ∈ h

−1

∞

(h(x))) = P

∞

(x).

42 2. Existence of a semiconjugation to a linear expanding endomorphism

Question 2.1 How can transitivity be recovered from the partition?

Lemma 8 If the set of points {x : P

∞

(x) = {x}} is dense in T

, then f is transitive.

Proof. Let us suppose that {x : P

∞

(x) = {x}} is a dense set. So, it is enough

to show that given x in t hat set, P

(x) t he element of the partitio n that contain it

for ﬁnitely many iterates by f cover T

. Observe that f

(x)) = P

(x)). Then

n+1

(x)) = f(P

(x))) = T

Remark 2.5 Actually, if the set of points {x : P

∞

(x) = {x}} is dense in T

, then f

is locally eventually onto. In fact, given V an open set, there exists z ∈ {x : P

∞

(x) =

{x}} ∩ V. So, P

(z) ∩ V = ∅. Taking n large enough we get tha t P

(z) ⊂ V. Therefore,

n+1

(V ) = T

Question 2.2 In the case that f is isotopic to a linear expanding endomorphism E. If

f is transitive, does it imply that the pre-orbit of any point by f are dense? does it imply

that the set {x : P

∞

(x) = {x}} is dense in T

? Do they have total (Lebesgue) measure?

Question 2.3 Knowing that there exists an expanding locally maximal set Λ with

certain geometrical properties such as the ones given by the Main Theorem. What can it

be said about the Markov Partition?

Deﬁnition 2.3 (Physical measure)

An invariant Borel probability measure µ for a dynamical system f : M → M is said

to be a physical or Sinai-Ruelle-Bowen(SRB) measure if it s basin of attra ction,

B(µ) = B( µ; f) := {z ∈ M :

n−1



i=0

(z)

→ µ weakly* as n → ∞},

has positive Lebesgue measure in M.

§ 2.2 The Case f is Isotopic to a Linear Expanding Endomorphism E 43

Question 2.4 Does there exist physical measures? If there exists, is it unique?

Question 2.5 Since f is semiconjugated to E, by the Bowen’s formula, see [Bow71], we

get that the topological entropy of f is bounded by below by the topological entropy of the

linear map. There a r e examples, such as Derived from an expanding linear endomorphism

where the central direction is one dimensional, the to pological entropy of the original map

and the semiconjugated are equal. Are there examples such as the topological entropy of

the semiconjugated map is greater than the initial map’s entropy?

Remark 2.6 The linear expanding maps preserve volume, they are Lebesgue inva-

riant.

So, this make us ask the following:

Question 2.6 Is it possible to construct a map f satisfying the hypotheses of the

Main Theorem and being volume preserving?

Remark 2.7 If f satisfy the hypotheses of the Main Theorem and assuming that f

is C

and volume preserving, then f is ergodic respect to Lebesgue measure.

44 2. Existence of a semiconjugation to a linear expanding endomorphism

Chapter 3

Exis tence of Robust Transitive

Endomorphism s

In this chapter we show that there exist examples of robust transitive endomorphisms

verifying the hypotheses o f our main results.

§ 3.1 Example 1: Applying Main Theorem

Consider E : T

→ T

an expanding endomorphism, with n ≥ 2. Note that taking a

large m > 1 , E

has the topological degree to the power m − th elements in the Markov

Partition, so without loss of generality we may assume that the initial map has many

elements in t he partition. More precisely, if N = deg(E), we may assume that N is large

and ther efore the Markov partition has N elements. Denote by R

the elements of the

partition, with 1 ≤ i ≤ N; R

is closed, int(R

) is nonempty and int(R

) ∩ int(R

) = ∅ if

i = j.

Now, consider ψ : T

→ T

a map isotopic to the identity and denote by



= ψ(R

)

for every i. The idea o f using this map is to deform the elements of the Markov partition

and get a new partition which elements are not all of the same size, there could be some

46 3. Existence of Robust Transitive Endomorphisms

very small, others very big.

Set U

an open set in T

such that if



U is the convex hull of the lift of U

, then



U ∩[0, 1]

is cont ained in the interior of [0, 1]

, i.e. diam

ext

) < 1. Note that there exist



such

that



∩U

is nonempty. We ask one more condition for U

, there are many



contained

in U

, this condition is feasible since we asked for the initial map to have many elements

in the partition.



Figure 3.1: Deforming the initial Markov Partition

Deﬁne f

: T

→ T

by f

= ψ ◦ E. We assume that there exist p ∈ U

and q

∈ U

ﬁxed points for f

, with 1 ≤ i ≤ n − 1, this is possible because we may start with an

expanding map which has as many ﬁxed points as we need.

Let us suppose that p and q

are expanding by f

in all directions, it means that all

the eigenvalues associated to t hese point s are in modulus greater than 1. Pick ε > 0 small

enough such that B

) ∩ U

= ∅ and B

) ∩ B

) = ∅ if i = j.

Let us denote the decomposition of the tangent space as follows

) = E

≺ E

≺ ··· ≺ E

n−1

≺ E

where ≺ denotes that E

dominates the expanding behavior of E

i−1

Next we deform f

by a smooth isotopy support ed in U

∪ (



)) in such a way

that:

§ 3.1 Example 1: Applying Main Theorem 47

1. The continuation of p goes through a pitchfork bifurcation, appearing two periodic

points r

, r

, such that both are repeller and p becomes a saddle point. But the new

map f still expand volume in U

2. Two expanding eigenvalues of q

become complex expanding eigenvalues. More pre-

cisely, we mix the two expanding subbundles of T

) corresponding to E

) and

i+1

), obtaining T

) = E

≺ E

≺ ··· ≺ F

≺ E

n−1

≺ E

, where F

is two

dimensional and correspond to the complex eigenvalues.

3. Outside U

∪ (



)), f coincides with f

4. f is expanding in U

5. There exists σ > 1 such that |det(Df( x))| > σ for every x ∈ T

Figure 3.2: f isotopic to f

§ 3.1.1 Property of Large Arcs

Claim 3.1 Every large arc in U

has a point such that its forward orbits remain in

Proof. Take d t he maximum of the diameter of the elements of the par t it ion contained

in U

. Note that every a r c in U

with diameter lar ger t han d cannot be contained in the

48 3. Existence of Robust Transitive Endomorphisms

interior of any element of the partition, more precisely has to intersect at least two elements

of the partition. Hence, the image by f of this a rc γ has diameter 1. So there exists a

piece of f (γ) in U

intersecting at least o ne element of the partition across two parallel

sides, let us call γ

. Choose a pre-image of γ

in γ a nd call it γ

Repeating the process for γ

, we have that there is γ

a piece of f(γ

) verifying t he

same condition as γ

. Then, choose γ

a pre-image of γ

by f

in γ.

Thus, we construct a sequence of nested arcs in γ. The intersection is non empty, a

point in this intersection satisfy our claim.

§ 3.1.2 Remarks About Example 1

1. q

’s ar e ﬁxed points for f with complex expanding eigenvalues. Note that the exis-

tence of these points ensures that the ta ngent bundle does not admit any invariant

subbundle. We could also start with an expanding map having, besides p, periodic

points q

with complex eigenvalues. In such a case, it is enough to make p goes

through a pitchfork bifurcation.

2. This example shows that U

can be as big as we desired while it veriﬁes the hypothesis

of having external diameter less than 1.

3. It can be constructed in any dimension.

§ 3.2 Example 2: Applying the Main Theorem Revisited

Let us consider E : T

→ T

an expanding endomorphism, with n ≥ 2. Assume that the

initial map has many elements in the Markov partition, let us say N element s.

Denote by R

the elements of the partition, with 1 ≤ i ≤ N. Since E is expanding, R

§ 3.2 Example 2: Applying the Main Theorem Revisited 49

are closed, int(R

) are no nempty and int(R

) ∩int(R

) = ∅ if i = j. Choose ﬁnitely many

of these elements, {R

}

j=1

, such that R

∩ R

= ∅ if i

= i

, i.e. they are two by two

disjoints. Consider the pre-images of every R

, let us say E

−1

) = {P

}

l=1

. Denote by

= R

. Next, we keep P

such that P

∩ P

= ∅ whenever 0 ≤ r = l ≤ N and i

= i

Finally, let us denote by {P

}

the collection of these latter subsets, so they are two by

two disjoint s.

Figure 3.3: {P

}

collec tion

Now, consider ψ : T

→ T

a map isotopic to the identity and denote by



= ψ(P

)

for every i.

Choose



an open connected subset such that its closure is contained in the interior



. Let φ

: T

→ T

be a map isotopic to the identity such that

• φ



is not expanding.

50 3. Existence of Robust Transitive Endomorphisms

• φ



is the identity.

Deﬁne φ : T

→ T

φ(x) =







(x), if x ∈



x, if x ∈





Hence, φ is equal to the identity in [





]

, expands volume but is not expanding in





Once we have deﬁned all these maps, we consider the map f = φ ◦ψ ◦E from T

onto

itself and denote by U

= int(





). Observe tha t f veriﬁes that:

(i) f is a volume expanding endomorphism.

(ii) f is an expanding ma p in U

(iii) Λ



n≥0

−n

) is an expanding locally maximal set for f which has the property

that separate large nice cylinders.

Since (i) and (ii) are immediate fr om the construction of f, we concentrate our interest

in to prove (iii).

§ 3.2.1 Λ

Separates Large Nice Cylinders

Note that by the construction of U

, we have that t he elements of the pre-orbit of U

are

two by two disjoints. Let us consider d

= max{diam

ext

(c.c.



n≥0

−n

))}. Since the

deﬁnition of U

, 0 < d

< 1.

§ 3.2 Example 2: Applying the Main Theorem Revisited 51

Claim 3.2 If γ is an arc in U

with diameter 1, then γ intersects Λ

Proof. Let γ be an arc in U

such that diam(γ) = 1. Suppose that γ does not intersect

Remember that Λ

= T



n≥0

−n

), it means that if x ∈ Λ

, then f

(x) ∈ U

Therefore, γ is contained in one pre-image of U

or in a union of pre-images of U

Observe that γ cannot be contained in just one pre-image of U

, because if it is con-

tained in f

−n

) for some n ≥ 0, then diam(γ) < diam(f

−n

)) < d

, which is absurd

because d

< 1.

Hence, γ should be contained in a union of pre-images of U

, since γ is compact we can

cover with a ﬁnite union of pre-images of U

. But we know that the pre- ima ges of U

are

two by two disjoints, hence there exist points in γ that cannot be covers by the pre-images

of U

. In particular, γ intersects Λ

Figure 3.4: Λ

looks like a Sierpinski set

Remark 3.1 We have a lready the existence of the invariant expanding locally max-

imal set Λ

. Moreover, by claim (3 .2) we get that this invariant set intersects every arc

52 3. Existence of Robust Transitive Endomorphisms

with large diameter. Then by the Main Theorem Revisited follows that this map is r obust

transitive.

§ 3.2.2 Remarks About Example 2

1. We can apply our Main Theorem R evisited to this example, obtaining in particular

that f is robustly tr ansitive.

2. The



’s can be as many and as big as we want.

3. We can construct ma ny examples starting with this initial map. In particular, we

can construct examples without invariant subbundles, such as putt ing a ﬁx point in

the complement of the U

with complex eigenvalues and doing a derived from an

expanding endomorphisms inside of some



§ 3.3 Example 3: A pplying Theorem 2

The idea of next example is to build an endomorphism in the 2-Torus which is a skew-

product and in the dynamic there is a blender. This example is more or less a standard

adaptation for endomorphisms of examples obtained in [BD96] for diﬀeomorphisms.

First, let us establish some notation before deﬁning the map. Pick 0 < a < 1/2 < b < 1

and denote I

= [a, 1] and I

= [0, b]. Note that I

∩ I

= [a, b]. This decomposition is

associated to the horizontal ﬁbers.

Next, ﬁx N > 3 and pick 0 < a

< b

< a

< b

< a

< b

< a

< b

< 1 such that

−a

= 1/N. Let us denote by I

= [a

, b

] with 1 ≤ i ≤ 4. Note that they are two by two

disjoint and do not contain 0 or 1. We associate this decomposition t o the vertical ﬁbers.

Let us call R

= I

× I

with i = 1, 2, and R

= I

× I

with i = 3, 4.

§ 3.3 Example 3: Applying Theorem 2 53

Now, deﬁne Φ : T

→ T

Φ(x, y) = (ϕ

(x), E(y)) ,

where ϕ

, E : S

→ S

are deﬁned as follows:

1. E is an expanding endomorphism such that:

• E(I

) = [0, 1] for every i.

• There exist a

< c

< b

such that E(c

) = c

2. ϕ

is deﬁned by ϕ

(x) = f

(x), if y ∈ I

, where f

: S

→ S

are diﬀerentiable maps

deﬁned as follows:

• f

and f

satisfy the following properties:

(i) f

has two ﬁxed points, 0 and a. 0 is a repeller and a is an attr actor for f

(ii) f

has two ﬁxed points, 0 and a

′

, where 0 < a

′

< a. 0 is an attractor and

′

is a repeller for f

(iii) f

) overlaps f

) and f

) ∪ f

) = I

(iv) |f

′

| < 1 and |f

′

| < 1.

• f

and f

satisfy the following properties:

(i’) f

has two ﬁxed points, 0 and b

′

, where b < b

′

< 1. 0 is a repeller and b

′

an attractor for f

(ii’) f

has two ﬁxed points, b and 1. b is an attractor and 1 is a repeller for f

(iii’) f

) overlaps f

) and f

) ∪ f

) = I

(iv’) |f

′

| < 1 and |f

′

| < 1.

3. |det(Df)| = |

∂ϕ

∂x

′

| > 1.

4. E

′

≫

∂ϕ

∂y

54 3. Existence of Robust Transitive Endomorphisms

Hence, the horizontal ﬁbers F

= S

×c

are invariant by f. Moreover, by condition (4),

the image of every vertical ﬁber is almo st a vertical ﬁber, in the sense that the tangent

vector is close to a vertical one; more precisely, the unstable cones family are almost

vertical.

Next, we consider Λ



n≥0

−n

∪ R

) and Λ



n≥0

−n

∪ R

). Let Λ



n∈Z

−n

∪ R

) and Λ



n∈Z

−n

∪ R

), note that both sets are expanding

invariant locally maximal sets.

§ 3.3.1 Λ

and Λ

Separate Large Vertical Segments

Let us denote by ℓ

(p) the vertical segment passing through p and length 1.

Claim 3.3 For every p ∈ R

∪ R

, follows tha t ℓ

(p) ∩ Λ

= ∅.

Proof. Let us suppose tha t p ∈ R

∪ R

. Note that for i = 1, 2, the imag e o f L

ℓ

(p) ∩R

by Φ has length 1 and by property (4) of Φ follows that Φ(L

) is almost vertical.

Moreover, L

∩F

= ∅ and Φ(L

∩F

) ∈ F

⊂ R

with i = 1, 2. Then Φ(ℓ

(p))∩(R

∪R

) = ∅.

Take K

one connected component of Φ(ℓ

(p)) ∩ R

, for i = 1, 2, such that P

) = I

where P

is the projection in the second coordinate. Consider the pre-image of K

by Φ

in L

and call it S

Now, iterate K

by Φ, doing a similar process we obtain K

a connected component

of Φ(K

) ∩ R

such that P

) = I

. Again take a pre-image of K

by Φ

, giving a

compact segment S

⊂ S

. Repeating this process, we may construct a nested sequence

of compact segment {S

}

in each R

. Thus,



is not empty and belong to ℓ

(p)∩Λ

Claim 3.4 For every p ∈ R

∪ R

, follows tha t ℓ

(p) ∩ Λ

= ∅.

Proof. By claim (3.3), we know that there exist a point z ∈ ℓ

(p) ∩ Λ

, this means

§ 3.3 Example 3: Applying Theorem 2 55

that Φ

(z) ∈ R

∪ R

for every n ≥ 0.

Then, just remain to show that there exist a sequence {z

}

k≥0

⊂ R

∪ R

such that

= z and Φ(z

) = z

k−1

. The idea o f the construction of such a sequence is to use now

the property (2-iii) of overlapping in the ho rizontal dynamics.

Knowing that Φ(R

) = f

) × [0, 1] and Φ(R

) = f

) × [0, 1], since property

(2-iii) we get that Φ(R

) ∩ Φ(R

) = f

) × [0, 1]. Hence, z

∈ (R

∪ R

) ∩ Φ(R

) or

∈ (R

∪ R

) ∩ Φ(R

), then there exists z

∈ R

∪ R

such that Φ(z

) = z

. Repeating

this process we construct the requires sequence.

Claim 3.5 For every p ∈ R

∪ R

, follows tha t ℓ

(p) ∩ Λ

= ∅.

Proof. The proof is similar to claim (3.4) just making the necessary arrangement.

Claim 3.6 For every q ∈ T

, we have that either ℓ

(q) ∩ Λ

= ∅ or ℓ

(q) ∩ Λ

= ∅.

Proof. Given any point q ∈ T

, note that ℓ

(q) ∩ R

= ∅ for every 1 ≤ i ≤ 4. Hence,

taking p

∈ ℓ

(q) ∩ R

and noting that ℓ

) = ℓ

(q), we may use claim (3.4) or (3.5) to

conclude that either ℓ

(q) ∩ Λ

= ∅ or ℓ

(q) ∩ Λ

= ∅.

§ 3.3.2 Remarks About Example 3

This example was constructed in the 2 -Torus with one dimensional centr al bundle, but we

can construct it in any T

and the dimension of the central bundle not need to be 1. Also,

we can use more than 4 dynamics in the horizontal, more precisely we put 2 blenders in

the dynamic but we can consider as many blenders as we want.

56 3. Existence of Robust Transitive Endomorphisms

§ 3.4 Example 4: A pplying Theorem 2

Let B

be an open ball in T

centered at 0 with r adius α < 1 and ϕ

: T

→ T

be a

diﬀerentiable map isotopic to the identity such that:

• ϕ

(0) = 0

• There exist 0 < λ

< λ

< 1 such that λ

< m{Dϕ

} < |Dϕ

| < λ

, i.e. ϕ

is a

contraction in a disk.

Let us consider D

the lift of B

to R

and ϕ

the lift of ϕ

. Note tha t ϕ

(0) = 0 a nd

< m{D ϕ

} < |D ϕ

| < λ

. By Proposition 2.3 of Nassiri’s PhD Thesis [Nas06],

there exists k ∈ N such that for every small ε > 0, there exist c

, . . . , c

∈ B

(0) and

δ > 0 such that B

(0) ⊂ Orbit

(0), where G = G( ϕ

, ϕ

+ c

, . . . , ϕ

+ c

) and Orbit

(0)

is the set of po ints lying on some orbit of 0 under the iterated function system (IFS)

G; mo re precisely, if we denote by



= ϕ

and



= ϕ

+ c

for i = 1, . . . , k, then

Orbit

(0) is the set of sequence {



(0)}

∞

l=1

where Σ

= (σ

, . . . , σ



◦ ··· ◦



and {σ

}

i∈N

∈ {0, . . . , k}

. (For more details about IFS see Nassiri’s PhD Thesis)

Now choose p

, . . . , p

∈ T

such that T

⊂



If φ

is the projection o f



on T

, deﬁne for every j the IFS G

= G

(φ

+ p

, φ

, . . . , φ

+ p

). Then B

) ⊂ Orbit

(0). Therefore, there exists an open set D

⊂ B

such that



) ⊃ D

, i.e. the IFS has the covering property. Hence,



′

)) ⊃

′

), with 0 < δ

′

≤ δ. Moreover, G

has also the overlapping property as in Example 3,

in the previous section.

Deﬁne the skew-product F : T

× T

→ T

× T

F (x, y) = (ψ

(x), E(y)) ,

where:

§ 3.4 Example 4: Applying Theorem 2 57

• E : T

→ T

is an expanding map with (k + 1)r ﬁxed points, let us denote the ﬁxed

points by e

, . . . , e

with 0 ≤ i ≤ k.

• For every y ∈ T

, ψ

: T

→ T

is a diﬀerentiable map isotopic to the ident ity such

that ψ

= φ

+ p

, with 0 ≤ i ≤ k and 1 ≤ j ≤ r.

Hence, every ﬁber T

× {e

} is invariant by F. Set R

= B

′

) × Q

, where Q

is a

small neighborhood of e

in T

such that E(Q

) = T

and they are all disjoints for every

i, j. Note that R

are the analogous of R

in the previous example.

Let Λ



n∈Z

(



i,j

§ 3.4.1 Λ

Separate Large Unstable Discs

Claim 3.7 Λ

veriﬁes that for every z ∈



i,j

follows that ℓ

(z) ∩ Λ

= ∅, where

ℓ

(z) is an unstable disc of internal diamet er 1 passing through z.

Proof. We may prove that there exists a point z ∈



i,j

such that F

(z) ∈



i,j

for every n ≥ 0 in a similar way as we proved claim (3.3) in previous example.

Moreover, for this z there exist z

∈



i,j

such that F (z

) = z. In fact, the idea is

more or less the same as in previous example, we must note that F (R

) = ψ

′

)) ×

E(Q

) = φ

′

)) × T

On the other ha nd, using the property of covering and overlapping follows that



i,j



i,j

′

) × Q

⊂



i,j

′

)) × E(Q

) = F (



i,j

Therefore, since z ∈



i,j

, there exist z



i,j

such that F (z

) = z. Inductively we can

construct a sequence {z

}

k≥0

⊂



i,j

such that z

= z and F (z

) = z

k−1

Thus, z ∈ ℓ

(z) ∩ Λ

58 3. Existence of Robust Transitive Endomorphisms

§ 3.4.2 Remarks About Example 4

This example is a generalization of Example 3. The intention here is to show that we may

apply Theorem 2 without taking into account the dimension of the central bundle and

this could be as large a s we want. Another observation is that the existence of blenders

guarant ee tha t our examples are robust transitive and this example veriﬁes the property

over the unstable discs with suﬃciently large internal diameter intersecting the invariant

expanding locally maximal set for the skew-product.

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