New classes of C 1 robustly transitive maps with persistent critical points
NNEW CLASSES OF C -ROBUSTLY TRANSITIVE MAPS WITHPERSISTENT CRITICAL POINTS Cristina Lizana
Departamento de Matem´atica. Instituto de Matem´atica e Estat´ıstica.Universidade Federal da Bahia. Av. Adhemar de Barros s/n, 40170-110. Salvador/Bahia, Brazil.
Wagner Ranter
Departamento de Matem´atica. Universidade Federal de Alagoas.Campus A.S. Simoes s/n, 57072-090. Macei´o/Alagoas, Brazil.Postdoctoral fellow at Math Section of ICTP. Strada Costiera, 11. I-34151, Trieste, Italy.
Abstract.
We exhibit a new large class of C -open examples of robustlytransitive maps displaying persistent critical points in the homotopy class ofexpanding endomorphisms acting on the two dimensional torus and the Kleinbottle. Introduction
A map is C -robustly transitive if there is a dense forward orbit, that is transi-tivity , for every map in a C -neighborhood. This issue has been focus of attentionby several authors.For diffeomorphisms, it is well-known that the existence of C -robustly tran-sitive maps requires some weak hyperbolicity structure. For further details see[11, 5], and [2]. It is widely known that in dimension two this structure implytopological obstructions. For instance, the only surface supporting a C -robustlytransitive diffeomorphism is the torus, and it must be homotopic to a hyperboliclinear diffeomorphism.For non-invertible maps ( endomorphisms ), on one hand [9] showed that any weakform of hyperbolicity is needed for the existence of C -robustly transitive local dif-feomorphisms (covering maps). On the other hand, we exhibit in [10] some topolog-ical obstructions for the existence of C -robustly transitive surface endomorphismsdisplaying critical points , that is points such that the derivative is not surjective.More precisely, it was proved in [10] that if F is a robustly transitive surface endo-morphism displaying critical points, then F has some weak form of hiperbolicity,namely partially hyperbolic , which, roughly speaking, means that there exists afamily of cone field on the tangent bundle which is preserved and its vectors are ex-panded by the derivative. Consequently, one has that the only surfaces supporting C -robustly transitive endomorphisms are the Torus and the Klein bottle. Further-more, it is shown that the action of every C -robustly transitive endomorphism onthe first homology group has at least one eigenvalue of modulus greater than one.In other words, every C -robustly transitive surface endomorphism is homotopicto a linear endomorphisms having at least one eigenvalue of modulus greater than AMS classification: 37D30, 37D20, 08A35, 35B38. Key words: robustly transitive endomor-phisms, critical set. a r X i v : . [ m a t h . D S ] A ug NEW CLASSES OF C -ROBUSTLY TRANSITIVE one. In particular, there are not C -robustly transitive endomorphisms homotopicto the identity. Hence, some natural questions arise. Question 1.
What are the homotopy classes admitting robustly transitive endo-morphisms?
Since some examples have been constructed recently in the 2-torus, anotherquestion is the following.
Question 2.
Does the Klein bottle admit robustly transitive maps?
Before doing some comments about the state of the art related to these questions,let us fix some notations. Throughout this paper End ( M ) denotes the set of allendomorphisms over M, where M is either the torus T or the Klein bottle K ,and the set of all critical points of an endomorphism F will be denoted by Cr( F ).Consider the linear endomorphism induced by the matrix L with all integers entries,which, by slight abuse of notation, we denote also by L , and µ, λ ∈ R its eigenvalues.For endomorphisms in End ( T ) without critical points, besides the classicalexamples as linear hyperbolic endomorphisms and expanding endomorphisms whichhave some hyperbolic structures, some examples appear in [12, 6] and [9]. However,these examples cannot be extended to obtain robustly transitive endomorphismsdisplaying critical points. The problem is, up to a perturbed, under the existenceof critical points, some open sets are sent onto a curve; and the classical argumentfor proving the robust transitivity via open set, that is given two open set there isan iterate of one of them intersecting the other one, could failed in the C -topologyunder the existence of critical points.The first example of robustly transitive endomorphisms displaying critical pointswas given in [4], this example is homotopic to a hyperbolic linear endomorphism,that is, it is in the homotopy class of L with | µ | < < | λ | . Later, [7] exhibitednew classes of examples which are in the homotopy class of a linear expandingendomorphism L . Unfortunately, there is a mistake in the proof for showing therobust transitivity of the constructed map in the homology class of L with 1 < | µ | < | λ | . Let us be more precise. In the proof of Proposition 2.1 in [7], theystart with a linear endomorphism L with 1 < | µ | < | λ | and obtain F ∈ End ( T )a perturbation C -close to L , but C -far from L , having persistent critical pointsand preserving the unstable cone field naturally defined for L . However, in orderto prove the robust transitivity of F , they use the fact that all images of any openset by F has nonempty interior, which is not always satisfied for endomorphismsdisplaying critical points as we commented above.One of the goal of this paper is to be a corrigendum of [7] and come to providethat such examples can in fact be constructed. Our approach is slightly bit differentfrom the one used in [7]. Let us state the first main result, for this consider L alinear endomorphism defined on the torus. Theorem A.
Let L be a linear endomorphism whose eigenvalues are µ, λ ∈ R sothat < | µ | (cid:28) | λ | . Then there are C -robustly transitive endomorphisms homotopicto L displaying critical points that are persistent under small perturbations. The condition 1 < | µ | (cid:28) | λ | means that | λ | is larger enough than | µ | such thatthe correspondent eigenspaces generate a dominated splitting . This will be usefulto create a region with a “good mixing” property which is persistent under smallperturbations, so-called “blender”. EW CLASSES OF C -ROBUSTLY TRANSITIVE 3 The proof is divided in two parts. The first part corresponds to the case theeigenvalues λ and µ are integer numbers, which we call Periodic case, see Section 2.1.The second one corresponds to the case the eigenvalues are irrational, so-calledIrrational case, see Section 2.2.Roughly speaking the idea of the construction is as follows. Start with an ex-panding linear endomorphism L with 1 < | µ | (cid:28) | λ | . Make a deformation in theweaker direction in order to get a “blender”. The new homotopic map is robustlytransitive endomorphism. Then, create artificially critical points far away from the“blender”, which are persistent under small C -perturbations, in such a way thatthe transitivity.The blenders were first introduced in [1] as a mechanism to create robustly tran-sitive non-hyperbolic diffeomorphisms. Blenders were used in [6] and [9] to createrobustly transitive non-hyperbolic local diffeomorphisms on the 2-torus. The def-inition of blender given in [6] is quite technical. There are several authors usingthe knowledge of blender in the non-invertible setting, for instance see [3]. We areassuming the notion of blender as follows, which is an adaptation of the definitionintroduced in [6], doing the consideration that for us the stable and unstable di-rections are on the contrary as it is in [6], that is, for us the stable direction ishorizontally and the unstable direction is vertical.Let I, J two closed intervals in S . Consider a rectangle Q = I × J in T and F : Q → T a C map. ( Q , F ) is a blender if there are two compact subsets R and R of Q such that F | R i is a diffeomorphism onto its image, for i = 1 ,
2, and
Q ⊆ F ( R ) ∪ F ( R ) verifying:(B1) F is hyperbolic on R i , with i = 1 ,
2. That is, denoting by E s and E u thetangent bundle to I and J , respectively, for all ( x, y ) ∈ Q one has that theunstable cone field C uα ( x, y ) = { u + v ∈ E s ( x, y ) ⊕ E u ( x, y ) : (cid:107) u (cid:107) ≤ α (cid:107) v (cid:107)} , for α > i ) DF ( C uα ( x, y )) ⊆ int( C uα ( F ( x, y )));( ii ) T ( x,y ) T = E s ( x, y ) ⊕ C uα ( x, y );( iii ) there exists 1 < σ < λ such that (cid:107) DF ( u ) (cid:107) < σ − (cid:107) u (cid:107) and (cid:107) DF ( w ) (cid:107) ≥ σ (cid:107) w (cid:107) , (1)for all u ∈ E s ( x, y ) and w ∈ C uα ( x, y );(B2) there exist two u-arc γ and γ (that is, γ (cid:48) i is contained in C uα ) of Q suchthat F ( γ ∩ R i ) ⊇ γ i and γ is strictly on one side of γ ;(B3) let V be the closed subset of Q between γ and γ , V a closed subset of Q\ V such that γ is part of the boundary of V and R (cid:48) i = ( R i ∩ V ) ∪ ( R i ∩ V )subset of Q , for i = 1 , , then F ( R (cid:48) i ) contains V i , i = 1 , F ( γ ∩ R ) ⊇ γ , there exists a hyperbolic saddle fixed point p F ∈ γ . In [6, Proposition 3.2] is proved that there exists a neighborhood U F of F so that ( Q , G ) is a blender for every G ∈ U F . Furthermore, the closure ofthe unstable manifold W u ( p G , G ) = ∪ n ≥ G n ( W uloc ( p G , G )) has nonempty interior,where p G is the continuation of p F .From the proof of Theorem A, we are able to construct on the Klein bottle a C -robustly transitive endomorphism having critical points that are persistent under NEW CLASSES OF C -ROBUSTLY TRANSITIVE small perturbations, answering Question 2 above affirmatively getting the secondgoal of this work.Before state next result, let us make some comments about the constructionof the examples on the Klein bottle. Let α, β : R → R defined by α ( x, y ) =( x + 1 , y ) , β ( x, y ) = ( − x, y + 1) , and Γ be the group of self-homeomorphisms of R generated by α and β . The Klein bottle is defined as the quotient space given by K = Γ \ R . Consider the diagonal matrix L = diag( µ, λ ), where µ, λ are nonzerointegers and λ is odd. This matrix induces a linear endomorphism on the Kleinbottle, so-called L , given by L [ x, y ] = [ µx, λy ] , ∀ [ x, y ] ∈ K . For further details see[8].We now are able to state the result. Theorem B.
Suppose that < | µ | (cid:28) | λ | . Then, there is C -robustly transitiveendomorphism on K displaying (persistent) critical points homotopic to a linearendomorphisms L . The proof is omitted since it is basically the same as for the torus case, seeSection 2.1. We emphasize that our approach requires a strong dominated conditionwhich is provided by the assumption | µ | (cid:28) | λ | . Some cases remain open as follows. Question 3.
Are there examples of C -robustly transitive endomorphism displayingcritical points in the homotopy class of a homothety? Question 4.
Can a zero degree endomorphism be C -robustly transitive? We believe that both questions can be answer affirmatively. For further discus-sions about this issue we suggest the readers to see [10]. The paper is organized asfollows. Section 1 is devoted to the basic notions and the prototype for the exam-ple. Section 2 is dedicated to the construction of the new large class of examplesatisfying Theorem A for both cases.1.
Preliminaries
Throughout this section, we introduce some essential notation and terminologiesthat will be used for proving Theorem A.1.1.
Iterated Function Systems - IFS.
Given f , . . . , f l : I → I orientationpreserving maps on an interval I , not necessarily invertible, we defined as IFS of f , . . . , f l the set of all possible finite compositions of f i ’s, that is, < f , . . . , f l > := { h = f i m ◦ · · · ◦ f i : i k ∈ { , . . . , l } , ≤ k ≤ m, and m ≥ } . The orbit of x is given by O ( x ) := { h ( x ) : h ∈ < f , . . . , f l > } , and the length | h | of the word h by m if h = f i m ◦ · · · ◦ f i .1.2. Prototypes.
By prototype, we mean a particular local feature of endomor-phism which exhibits a blender. In our approach, we will deform the linear endo-morphism in order to obtain locally one of the following prototypes.Suppose that F : T → T is an endomorphism admitting a box Q = I × J ⊂ T ,and two disjoint intervals J and J contained in J so that F ( x, y ) = ( f i ( x ) , λy (mod 1)) , ∀ ( x, y ) ∈ I × J i , satisfying the following conditions:(P.1) λ > λJ i := { λy : y ∈ J i } ⊃ J ; EW CLASSES OF C -ROBUSTLY TRANSITIVE 5 Figure 1.
Periodic case.
Figure 2.
Irrational case.(P.2) f , f : I → I are as one of the cases in Figures 1 and 2.We claim that ( Q , F ) is a blender according to the definition given in the In-troduction. In fact, for both cases (Figures 1 and 2), we have that F is hyper-bolic on R i = I × J i , with i = 1 ,
2, since there exists α > x, y ) ∈ R i , i = 1 , T ( x,y ) T = E s ( x, y ) ⊕ C uα ( x, y );(ii) DF ( x, y ) is the diagonal matrix diag( f i ( x ) , λ ) on E s ( x, y ) ⊕ E u ( x, y );(iii) there exists 1 < σ < λ such that: | f (cid:48) i ( x ) | < σ − and (cid:107) DF ( x,y ) ( w ) (cid:107) ≥ σ, ∀ w ∈ C uα ( x, y ) . (2)Then, item (B1) holds. Moreover, the u -arcs γ and γ defined as { x } × J and F ( { x } × J ), respectively, with x a fixed point for f , satisfy item (B2). Item(B3) follows easily from the construction. That concludes the claim.Recall that if ( Q , F ) is a blender then there is a saddle point p F such that theclosure of the unstable manifold W u ( p F ) of F has nonempty interior. Furthermore,there is a neighborhood U F of F in End ( T ) so that for every G ∈ U F , we can seethat the unstable manifold W u ( p G ) of G at the continuation point p G of p F alsohas nonempty interior. Remark 1.2.1.
Observe that the local stable manifold W sloc ( p F ) for F at the hy-perbolic fixed saddle point p F contains the interval I . Recall that the local stablemanifold W sloc ( p G , G ) depends continuously on G in U F . Construction of the examples
In this section, we are dedicated to prove Theorem A. The proof is split intotwo parts. The first part, it is the periodic case, when the eigenvalues µ and λ areintegers. The second one, it is when µ and λ are irrational. Recall that µ and λ are eigenvalues of a square matrix L with all integer entries.Let us fix some notation and terminologies. Write E c and E u to denote thesubspaces associated to the eigenvalues µ and λ , respectively. We denote F c and F u the foliations induced on the torus by the eigenspaces E c and E u , called (weak)unstable foliation and strong unstable foliation respectively. The leaves of F c and F u are preserved by the linear endomorphism L .We define for each α > p ∈ T the unstable cone at point p by: C uα ( p ) = { ( u, v ) ∈ E c ⊕ E u : (cid:107) u (cid:107) ≤ α (cid:107) v (cid:107)} . (3)It easy to see that for every α >
0, we have L ( C uα ) is contained in C uµα/λ . NEW CLASSES OF C -ROBUSTLY TRANSITIVE Periodic case.
We here assume that µ and λ are integers and that the mod-ulus of λ is sufficiently larger than the modulus of µ to provide the dominatedstructure. It easy to see that, in this case, the unstable and strong unstable folia-tions consist of closed curves. For the sake of simplicity, we may assume withoutloss of generality that the leaves of F c and F u are horizontal and vertical closedcurves respectively. That is, the leaves of F c and F u are the following closed curves: F c ( x, y ) = S × { y } and F u ( x, y ) = { x } × S , ∀ ( x, y ) ∈ T = R / Z . (4)2.1.1. One-dimensional dynamics.
Take f : S → S as f ( x ) = µx (mod 1). Forevery ε > f to obtain two maps f and f whichrestricted to the interval ( − ε, ε ) ⊆ S are contracting affine maps as in Figure 3. Figure 3.
The graphs of f and f with topological degree µ .Furthermore, the maps f and f are orientation preserving C -local diffeomor-phisms on S of degree µ and verify the following:(1) f i is 2 ε C -close to f ( x ) := µx (mod 1) with 0 < | ∂ x f i | < − ε, ε );(2) there exist points in S , − ε < p < < p < ε , where f i ’s restricted to I = [ p , p ] are affine maps and p i is an attractor fixed point of f i , i = 1 , f i has two repeller fixed points p i − and p i + in a neighborhood of the attractor p i satisfying p i − < − ε < p i < ε < p i + .Note that f and f restricted to I are as in Figure 1.(4) the map f i ( x ) restricted to S \ ( − ε, ε ) is an affine function of the form α i x + β i , where α i is sufficiently close to µ and β i is chosen such that itsforward orbit by f , O + ( β i , f ) = { f k ( β i ) : k ≥ } , is dense in S .We can assume ∂ x f i < µ (cid:48) , for some µ (cid:48) close to µ but µ (cid:48) > µ , and i = 1 , Two-dimensional dynamics.
We define F : T → T as a skew product map F ( x, y ) = ( f y ( x ) , λy (mod 1)), where f y : S → S is an homotopy map satisfying: f y ( x ) = (cid:26) f i ( x ) , ( x, y ) ∈ R i for all i = 0 , , f ( x ) , ( x, y ) ∈ T \R J , (5)where R J and R i are horizontal stripes in T of the form S × J and S × J i ,respectively; the interval J in S centered at the origin (the fixed point of f ), and J , J and J are three pairwise disjoint intervals contained in J sufficiently smallsuch that λJ i = { λx : x ∈ J i } ⊇ J . Then, we should verify that F restricted to thebox Q = I × J , where I = [ p , p ], is as the prototype in Figure 1 which impliesthat ( Q , F ) is a blender.These intervals J i can be easily chosen for large λ ( e.g., λ ≥ C -robustly transitive endomorphisms displaying critical points homotopic EW CLASSES OF C -ROBUSTLY TRANSITIVE 7 to expanding linear endomorphisms, but not interested to quantify exactly which λ should be.2.1.3. Weak hyperbolicity.
Here we show that the skew product map F defined inthe Section 2.1.2 has a hyperbolic structure. In order to prove that, we show that F preserves the unstable cone field C uα defined in (3).We need some control on the derivative respective to the second variable forconstructing an invariant family of cones. The following result shows that we maychoose the homotopy f y so that | ∂ y f y | is sufficiently small. Proposition 2.1.1.
Given δ > , there exists ε > such that for f, g : S → S C -maps of topological degree µ which are ε C -close, there exists an homotopy H t between f and g so that | ∂ t H t | < δ. Proof.
Let π : R → S be the universal covering map. Let ˜ f and ˜ g be the liftof f and g , respectively, such that ˜ f and ˜ g are ε C -close each other. Then, wemay define the isotopy from ˜ f to ˜ g by ˜ H t (˜ x ) = (1 − t ) ˜ f (˜ x ) + t ˜ g (˜ x ), for every˜ x ∈ R , t ∈ [0 , H t (˜ x + m ) = ˜ H t (˜ x ) + µm , for every ˜ x ∈ R , m ∈ Z , we havethat the homotopy H t ( x ) = π ◦ ˜ H t (˜ x ) between f and g is well defined. Furthermore, | ∂ t H t | ≤ (max ˜ x ∈ R | Dπ (˜ x ) | ) | ∂ t ˜ H t | . Since | ∂ t ˜ H t | = | ˜ f (˜ x ) − ˜ g (˜ x ) | < ε and max ˜ x ∈ R | Dπ (˜ x ) | is bounded, we may choose ε > | ∂ t H t | < δ . (cid:3) We now assume that f y is a homotopy that | ∂ y f y | < δ . Next, it will be specifiedsome restriction over δ > F . Lemma 2.1.1 (Unstable cone fields for F ) . Given δ > and α > , there exist < θ := θ ( δ, α ) < and < λ (cid:48) < λ such that the following properties hold: (a) DF p ( C uα ( p )) \{ (0 , } ⊂ C uθα ( F ( p )) , for every p ∈ T ; (b) if w ∈ C uα ( p ) , then (cid:107) DF p ( w ) (cid:107) ≥ λ (cid:48) (cid:107) w (cid:107) .Proof. The proof follows easily from the facts | ∂ y f y | < δ and the derivative of F is DF = (cid:18) ∂ x f y ∂ y f y λ (cid:19) with | ∂ x f y | + δ < λ . (cid:3) It is well known that an unstable cone field is shared by every endomorphism C -close enough to F . That is, there exists a neighborhood U F of F in End ( T ) sothat the unstable cone field C uα so is an unstable cone field for any G in U F . Thus,it follows direct from Lemma 2.1.1 the following statement. Corollary 2.1.
There exists < λ (cid:48) < λ such that for every G ∈ U F and everyu-arc γ (that is, γ (cid:48) ⊂ C uα ), holds that (cid:96) ( G n ( γ )) ≥ ( λ (cid:48) ) n (cid:96) ( γ ) for each n ≥ . Next statement guarantees that every arc tangent to the unstable cone field hassome iterate “almost” like the leaves of F ui which is the connected component ofthe unstable foliation F u in R i . Proposition 2.1.2.
For every τ > small enough, we can take U F small enoughsuch that for every G in U F and every u -arc γ (that is, γ (cid:48) ⊂ C uα ), there exist asequence of points ( q n ) n and a sequence of u -arc ( γ n ) n in R i satisfying: NEW CLASSES OF C -ROBUSTLY TRANSITIVE ( i ) q n ∈ γ n , γ ⊂ γ, and γ n ⊂ G ( γ n − ) ; ( ii ) ( γ n ) n is τ C -close to F ui ( q n ) for large n .Proof. First, note that the cone C uα is an unstable cone field for G and the curve G n ( γ ) grows up transversely to the weak unstable foliation F c . Then, up to get aniterate of γ , we can assume γ cross the stripe R i . Furthermore, we get γ n as thecomponent of G ( γ n ) in R i , where γ is the u -arc γ . Each curve γ n is an u -arc andintersects each leaf of F c in the stripe R i exactly once.Since F ( x, y ) = ( f i ( x ) , λy (mod 1)) in R i , we can shrink U F , if necessary, to getthat for every G = ( g , g ) ∈ U F holds that | ∂ x g | < µ (cid:48) , | ∂ y g | , | ∂ x g | < τ , and | ∂ y g − λ | < τ .Let v = ( v c , v u ) be an unit vector in C uα ( q ) with slope ρ = | v c | / | v u | . We take q n = G n ( q ) ∈ R i , for each n ≥
0, and v n = DG ( v n − ) ∈ C uα ( q n ) with slope ρ n .Then, we can see inductively that ρ n = | v cn || v un | = | ∂ x g ( q ) v cn − + ∂ y g ( q ) v un − || ∂ x g ( q ) v cn − + ∂ y g ( q ) v un − | ≤ µ (cid:48) ρ n − + τ | λ − τ | − τ ρ n − . (6)Since DG ( C uα ( q )) \{ (0 , } ⊂ C uα ( G ( q )), we have ρ n < α and ρ n ≤ µ (cid:48) ρ n − + τ | λ − τ | − τ ρ n − ≤ ρ n − b + τµ (cid:48) b where b = ( | λ − η | − τ α ) /µ (cid:48) >
1. Then, ρ n ≤ ρ b n + τµ (cid:48) n (cid:88) i =1 b i = ρ b n + τ (1 − b − n ) µ (cid:48) ( b − . By compactness, the vector v can be chosen so that ρ is the maximum possibleslope of unit vectors in C uα . This together with the fact that G ( (cid:107) γ (cid:48) n ( q ) (cid:107) ) ≥ λ (cid:48) (cid:107) γ (cid:48) n ( q ) (cid:107) guarantee that γ n is τ C -close to F i for large n . (cid:3) Dynamics properties for IFS.
Since F ( x, y ) = ( f i ( x ) , λy (mod 1)) for all( x, y ) ∈ R i , we can see that each strong unstable leaf F u ( x, y ) in R i is the verticalinterval { x } × J i , and so, F ( { x } × J i ) contains the vertical interval { f i ( x ) } × J which contains { f i ( x ) } × J i for each i = 0 , ,
2. Then, the orbit of the verticalinterval { x } × J i by F contains { h ( x ) } × J with h in IFS < f , f , f > .Thus, in order to understand the orbit of F we show here some dynamics prop-erties for IFS < f , f , f > . Proposition 2.1.3.
For every x ∈ S , the orbit O ( x ) by the IFS < f , f , f > intersects the interval I .Proof. Since p i ∈ ( p i − , p i + ) is the unique attractor for f i , for i = 1 ,
2, it is enough toshow that O ( x ) intersects ( p − , p ). Indeed, for each h ( x ) ∈ ( p − , p ), we can iterateit either by f or f , getting either f k ( h ( x )) close to the attractor p or f k ( h ( x ))close to p . Then, we iterate by either f or f , so that the orbit enters in I .We prove now that for each x on the circle, there is h in < f , f i > such that h ( x )belongs to ( p i − , p i + ). In order to prove that, we suppose without loss of generalitythat f k ( x ) / ∈ ( p i − , p i + ) for every k ≥
0. Then, we observe that f ◦ · · · ◦ f (cid:124) (cid:123)(cid:122) (cid:125) n − k ◦ f i ◦ f ◦ · · · ◦ f (cid:124) (cid:123)(cid:122) (cid:125) k − ( x ) = f n − k ( f i ( f k − ( x )) = µ n − α i x + f k ( β i ) , EW CLASSES OF C -ROBUSTLY TRANSITIVE 9 for 1 ≤ k ≤ n −
1, belong to the orbit O ( x ) = { h ( x ) : h ∈ < f , f , f > } . Recallthat f i ( x ) = α i x + β i for x ∈ S \ ( p i − , p i + ) and { f k ( β i ) : k ≥ } is dense in S . Thus,we can conclude that { h ( x ) : h ∈ < f , f i > } is dense in S . (cid:3) The following holds as an immediate consequence of Proposition 2.1.3 and theconstruction of F . Lemma 2.1.2.
For every x ∈ S , the closure of {{ h ( x ) } × J : h ∈ < f , f , f > } intersects the blender ( Q , F ) . Consequently, for every x ∈ S , it follows that Q ∩ (cid:0) ∪ k ≥ F k ( { x } × J ) (cid:1) (cid:54) = ∅ . (7)We know, by Remark 1.2.1, that the local stable manifold W sloc ( p F ) contains theinterval I . Then, the set ∪ k ≥ F k ( { x } × J ) intersects W sloc ( p F ) for every x ∈ S .By compactness, we can choose an iterate ∪ nj =0 F j ( { x } × J ) which intersects thelocal stable manifold W sloc ( p F ) for each x ∈ S . Finally, shrinking, if necessary, theneighborhood U F , we can see that ∪ nj =0 G j ( { x }× J ) is close enough to ∪ nj =0 F j ( { x }× J ) and, using that the local stable manifold W sloc ( p G ) depends continuously of G ,holds ∪ j ≥ G j ( { x } × J ) intersects W sloc ( p G ) for each G ∈ U F . Therefore, we canconclude the following. Lemma 2.1.3.
There exists a neighborhood U F of F such that for every G ∈ U F and every x ∈ S , we have that W sloc ( p G , G ) ∩ (cid:0) ∪ k ≥ G k ( { x } × J ) (cid:1) (cid:54) = ∅ . (8) More general, for each u -arc γ and every G ∈ U F , we have that the orbit ∪ n ≥ G n ( γ ) intersects W sloc ( p G ) Proof.
Recall that W sloc ( p G ) depends continuously on G and that W sloc ( P F ) containsthe interval I (which form the blender Q = I × J ). Since F n ( x, y ) convergesto p F for every ( x, y ) ∈ W sloc ( p F ), we can assume that { x } × J hits W sloc ( p G ).By compactness, we can take a large positive integer n such that ∪ nj =0 F j ( { x } × J i ) intersects W sloc ( p F ) for every x ∈ S and i = 0 , ,
2. In particular, the set ∪ nj =0 F j ( { x } × J i ) intersects W sloc ( p G ) for every G ∈ U F . Then, shrinking U F ifnecessary, we can assume that G j is close to F j for each j = 1 , . . . , n so that ∪ nj =0 G j ( { x } × J i ) also intersects W sloc ( p G ) which proves the first part of the lemma.We know that the length of G n ( γ ) grows exponentially and keep being tangentto the unstable cone C uα , for each u -arc γ . Then, we can assume that G n ( γ ) crossesthe stripe R i for each i = 0 , ,
2. Say γ a component of some G n ( γ ) in R i . Thus,up to shrink the neighborhood U F , we can take by Proposition2.1.2 a sequence( γ k ) k which γ k is the component of G ( γ k − ) in R i and γ k is “almost” vertical as { x } × J i for large k and some x ∈ S . Then, we can take k large enough such that G j ( γ k ) stay close to G j ( { x } × J i ) for several iterates, so that some iterated G j ( γ k )hits the local stable manifold W sloc ( p G ). (cid:3) Remark 2.1.1.
We would like to point out that the IFS < f , f , f > is used tobuild the blender for F and to guarantee that every local strong unstable leaf F uloc ( p ) (“vertical” interval) has some iterate that intersects the blender. In consequence,these properties are used to extend the same properties for all G sufficiently closeto F and for all u -arc γ . C -ROBUSTLY TRANSITIVE Additionally, note that as F is expanding outside of the blender and the closureof W u ( p G , G ) has nonempty interior for G nearby F , for further details see [6,Proposition 3.2]. We can assume, up to shrinking U F , the following statement. Lemma 2.1.4.
The image of the closure of W u ( p G , G ) by a large iterate of G isthe whole surface T , for every G ∈ U F . Creating critical points.
We make here a local perturbation to create artifi-cially the critical points of all endomorphisms in U F . Artificially means that thecritical points will be created in such a way that all the previous properties obtainedso far are preserved.Let B t,s be a box centered at the origin with sides of length t and s respectively,with t, s > C ∞ maps ψ : R → [0 , µ + ] and ϕ : R → R such as in Figure 4 verifying: • ψ ( x ) = ψ ( − x ) and µ < ψ (0) < µ + ; • ϕ (0) = 0 , min { ϕ (cid:48) } ≥ − λ − µµ +1 , and max { ϕ (cid:48) } = 1. Figure 4.
The graphs of ψ and ϕ (cid:48) , respectivelyGiven ( x , y ) ∈ T , we define F t,s : T → T by F t,s ( x, y ) = ( f y ( x ) − Φ( x, y ) , λy (mod 1)) , where Φ( x, y ) = ϕ ( x − x ) ψ ( y − y ) for every | x − x | < s , | y − y | < t ; andΦ( x, y ) = 0, otherwise. Adapting the arguments used in [7, Lemma 2.1.1] followsthat, for t, s small enough, F t,s still preserves the weak hyperbolicity structure, thatis the unstable cone family is preserved.Choose ( x , y ) ∈ T such that B t,s centered at ( x , y ) is away from the stripe R J , this ball is disjoint from the blender region and, in consequences, the previouslemmas hold for F t,s . Note that the critical points of F t,s are given byCr( F t,s ) = { ( x, y ) ∈ T : µ − ψ ( y − y ) ϕ (cid:48) ( x − x ) = 0 } . (9)Since F is a C -endomorphism having points on which the determinant of DF isnegative (e.g., in ( x , y )) and positive, we have persistence of the critical points inthe C -topology. Remark 2.1.2.
Note that | ϕ | goes to zero as s goes to zero. Hence, it holds that F t,s goes to F in the C topology, when t and s go to zero. In particular, since F was constructed C -close to L , it follows that F t,s is C -close to L . Hence, F t,s ishomotopic to L . From now on, by slight abuse of notation, we fix t, s small enough, and call F = F t,s . Finally, we prove that F is robustly transitive, and therefore, satisfiesTheorem A. The graph and conditions of ϕ (cid:48) are slightly different from the one in [7], since the one used in[7] does not satisfy the properties needed to preserve the weak hyperbolic structure. EW CLASSES OF C -ROBUSTLY TRANSITIVE 11 Proof of Theorem A for the periodic case.
Let U and V two open sets in T and G ∈ U F . By Lemma 2.1.3 and Lemma 2.1.4, there are an iterate of an u -arc γ contained in U which intersects W sloc ( p G , G ) and some iterate of the closureof W uloc ( p G , G ) which intersects V . Next, we take a “horizontal” arc β containedin some preimage of V that hits W uloc ( p G , G ) and is transversal to the unstablecone. Finally, since p G is a hyperbolic fixed point, we conclude that G n ( γ ) has acomponent γ n which is “almost vertical” and is getting close to W uloc ( p G , G ), hitting β . Therefore, G is transitive and this concludes the proof for the periodic case.2.2. Irrational Case.
From now on, we assume that the eigenvalues µ and λ areirrational with 1 < | µ | (cid:28) | λ | , recalling that 1 < | µ | (cid:28) | λ | means that | λ | is largerenough than | µ | such that the correspondent eigenspaces generate a dominatedsplitting. Let v c and v u be the unit eigenvector of L associated to the eigenvalues µ and λ , respectively. Recall that the eigenspaces E c and E u are generated by v c and v u and the leaf F σ ( q ) is the line { q + tv σ : t ∈ R } on T , for every q ∈ T and σ = c, u . Given q ∈ T , denote by F σ, ± ( q ) the connected components of F σ ( q ) \{ q } ,that is, F σ, + ( q ) is the semi-line { q + tv σ : t ≥ } , similarly for F σ, − ( q ) with t ≤ F σ, ± loc ( q ) as the components of F σloc ( q ) \{ q } .2.2.1. Sketch of the proof.
Since the proof is slightly more technical than the peri-odic case, we present here the strategy and main steps of the proof. However, thegeneral idea is the same as for the periodic case.
Step I:
First, we create a blender. Using the density of F u, + ( p ) , with p = (0 , , on the torus, we make a local deformation of L far away from p in order to createa cycle at p , let us call by F the perturbed map; that is, we find an interval I in F u, + loc ( p ) and a positive integer n such that F n ( I ) contains F u, + loc ( p ). Remark 2.2.1.
It should be noted that for the periodic case, the strong unstablemanifold F u ( x, y ) is the closed curve { x } × S . Hence, for the periodic case it isnot needed to make a deformation to create a cycle, since any interval I in F u ( x, y ) has an iterate by L which is the whole F u ( x, y ) . Then, we perturbe F in a small neighborhood of p where F coincide with L ,so that the expanding fixed point p of L becomes a hyperbolic saddle point. Thisnew map is called “ Derived from Anosov ” of F , DA for short. This perturbation isdone in such a way that the DA-endomorphism, which by abuse of notation we alsodenote by F , preserves the cycle created previously. In addition, it also preservesthe unstable cone field C uα defined in (3), and min (cid:107) DF ( v ) (cid:107) ≥ λ (cid:48) for some λ (cid:48) > v in C uα .Finally, we get a box Q = I × J where I is an interval in W sloc ( p, F ) and J is aninterval in W uloc ( p, F ) where p is a common extreme of I and J . Next, we choosean interval J ⊆ J containing p which F ( J ) ⊇ J ; and, using the existence of acycle for F , choose another interval J ⊆ J and n ∈ Z + such that F n ( J ) ⊇ J .Furthermore, we show that there is (cid:96) ∈ Z + such that F (cid:96) ( x, y ) = ( f i ( x ) , λ (cid:96) y (mod 1))on ( x, y ) ∈ R i = I × J i , where i = 1 , , and f , f : I → I are as Figure 2. Then,we conclude that ( Q , F ) is a blender and, therefore, a blender for every G closeenough to F in the C -topology. Step II:
After ensured the existence of a blender, we prove that every u -arc hassome iterate by F that intersect the blender. Then, we extend this property forevery endomorphism C -close to F , using the density of all the leaves of F u and the C -ROBUSTLY TRANSITIVE fact that F coincides with L out of the region of perturbation. The rest of the prooffollows from adapting the arguments of the periodic case to conclude Theorem Afor the irrational case. Remark 2.2.2.
Recall that in the periodic case the IFS < f , f , f > is used toguarantee that every u -arc has some iterate which intersect the blender. Step I: creating a blender.
We now prove the first step given in the previoussketch, recalling that p = (0 ,
0) is an expanding fixed point of L . Lemma 2.2.1 (Cycle) . For every neighborhood U of the identity in End ( T ) , thereexists h t ∈ U such that (a) F t := h t ◦ L is C -close to L ; (b) there exists I ⊂ W uloc ( p, F t ) \{ p } an interval such that F nt ( I ) contains thelocal unstable manifold W uloc ( p, F t ) .Proof. Observe that if there are a positive integer n and an interval I containedin F uloc ( p ) \{ p } such that L n ( I ) contains the local unstable manifold F uloc ( p ) of L ,then there is nothing to prove and L verifies the lemma. Hence, let us assume that L n ( F uloc ( p ) \{ p } ) does not contain p for every positive integer n .Let us prove item (a). Note that given q on the torus, we may rewrite L locallyas L ( x, y ) = q + ( µx, λy ) for every ( x, y ) on the rectangle I cr × I ur , where I σr is aninterval on E σ centered at the origin and radius r , for σ = c, u . Let ϕ : R → R bea bump function such that ϕ ( s ) = 1, if | s | ≤ /
4, and ϕ ( s ) = 0, if | s | ≥
1. Theprojection of the rectangle I cr × I ur centered at the origin defined on the tangentbundle, by slightly of abuse, is denoted by I cr × I ur , observing that now it is arectangle centered at q on the torus. Then, define h t : T → T as h t = Id outsideof the rectangle I cr × I ur centered at q ; and define h t on I cr × I ur as the followingmap, h t ( q + ( x, y )) = q + ( x + t Φ( x, y ) , y ) , ∀ ( x, y ) ∈ I cr × I ur , (10)where Φ( x, y ) = ϕ ( x/r ) ϕ ( y/r ). For simplicity, we will omit the point q in (10).Note that h t deforms the strong unstable leaf that crosses the rectangle on the v c -direction, and has the form h t ( x, y ) = ( x + tϕ ( x/r ) , y ) for every ( x, y ) ∈ I cr × I u r / .Denote by R the rectangle I cr × I u r / and notice that h t translates interval in R parallel to the strong unstable direction. Fix r and r small enough such that for0 < t < r , h t is C -close to the identity and F t = h t ◦ L is C -close to L . Itconcludes the proof of item (a).The local translation h t is supported in the region R that will be stablished inClaim II below. Claim I:
There exists a sequence ( I uj ) j of strong unstable intervals contained in F u, + ( p ) ∩ R with the following properties:(i) there exists ( n j ) j positive integers so that I uj ⊆ L n j ( I uj − ) for each j ≥ ≤ k < n j , the interval L k ( I uj − ) does not intersect R .Recall that any internal I contained in some strong unstable leaf of F u verifiesthat the length of L n ( I ) grows exponentially and got dense on the whole torus.Thus, we can take a positive integer n as the first time that L n ( F u, + loc ( p )) crossesthe rectangle R . Denote the component of L n ( F u, + loc ( p )) that crosses the rectangle R by I u . We now observe that L n ( I u ) also get dense as n increase. Again, we EW CLASSES OF C -ROBUSTLY TRANSITIVE 13 can choose n as the first time that L n ( I u ) crosses R ; and denote by I u suchcomponent. It is possible because L n ( I u ) is getting dense on the torus by parallelsegments and so if L n ( I u ) hits on R , but does not cross it, then we remove theintersection L n ( I u ) ∩ R and keep iterating L n ( I u ) \ ( L n ( I u ) ∩ R ) until it crossesthe rectangle R for the first time. Repeating that process indefinitely, we obtain asequence ( I uj ) j in R such that I uj ’s are contained in F u, + ( p ) and verify I uj containedin L n j ( I uj − ) for every j ≥
1. Furthermore, by the choices of n j ’s, we have that L k ( I uj ) does not hit the region R for each 1 ≤ k < n j . This concludes the claim. Claim II:
Given ε >
0, there is n ≥ I ⊆ F u, + loc ( p ) \{ p } so that F nt ( I ) ⊆ F uloc ( p ) containing p , for some 0 < t ≤ ε .Since F t coincides with L outside of the rectangle I cr × I ur , we get that W uloc ( p, F t ) = F uloc ( p ). Then, Claim II allows us to conclude the proof of item(b), and so, complete the proof of Lemma 2.2.1.Let us now fix the center of the rectangle I cr × I ur , that is fix q such that q = q + ( r ,
0) is the center of the vertical right boundary of the rectangle andit belongs to L − ( p ). By Claim I, consider a sequence of positive integers ( n j ) j and intervals ( I uj ) j contained in R such that I u is contained in L n ( F u, + loc ( p )) and I uj ⊆ L n j ( I uj − ) and L k ( I uj − ) does not hit on I cr × I ur for each 1 ≤ k < n j . Assumewithout loss of generality that I uj = L n j ( I u ) for some n j large enough so that µ n j ε ≥ r .We are now able to prove Claim II. Since, for each 0 < t ≤ ε , the map F t isa v c -translation on I cr × I ur and F t = L outside of the rectangle I cr × I ur , wehave that the image of each leaf F u ( x, y ) by F t is the leaf F c ( L ( x, y )) for every t .Moreover, we also have that there is I ⊆ F u, + ( p ) such that its n -iterate by F t isequal to h t ◦ L n ( I ) , that is a v c -translation of I u on R . We denote it by I t andtake the family R = { I t : 0 ≤ t ≤ ε } . Define R j as the set { F n j t ( I t ) : 0 ≤ t ≤ ε } .Then, using that F t preserves the weak unstable foliation F c for every t , we havethat R j is a family of intervals ( I jt ) for 0 ≤ t ≤ µ n j ε contained in R such that eachinterval I jt is equal to F n j t ( I t (cid:48) ) for some 0 ≤ t (cid:48) ≤ ε . In particular, I j is equal to I uj = L n j ( I u ).Therefore, we conclude that there is I ⊆ F u, + loc ( p ) and 0 < t ≤ ε such that F n + n j t ( I ) = F n j t ( I ) contains the interval { r } × I u r / , which implies that F nt ( I )is an interval in F uloc ( p ) containing p . This complete Claim II and, consequently,proves that F t has a cycle. (cid:3) Now we get a Derived from Anosov (DA) from F t in order to create a blender. Lemma 2.2.2 (Derived from Anosov) . Let F t be the C -map given by Lemma 2.2.1.There is a DA from F t , denoted by F , which has a blender for some iterated F n .Proof. Consider a small ball B centered at the fixed point p = (0 ,
0) and radius r >
0. Fix 0 < η < r , define ξ : R → [0 ,
1] a bump function such that ξ ( z ) = 0 in B c and ξ ( z ) = 1 for (cid:107) z (cid:107) ≤ η . Define the family of diffeomorphisms Θ s : T → T byΘ s ( x, y ) = ( xe − sξ (( x,y )) , y ) in the ( v c , v u )-coordinates, if ( x, y ) ∈ B , and Θ s is equalto the identity on T \ B . Then, the endomorphism F s,t given by Θ s ◦ F t satisfiesthe following properties: • p is a hyperbolic fixed point and the derivative of DF s,t at p in the ( v c , v u )-coordinates is a diagonal matrix DF s,t ( p ) = diag( e − s µ, λ ), denote e − s µ by µ s ; C -ROBUSTLY TRANSITIVE • the map F s,t = F t in T \ B which is an expanding map. Furthermore, as F t = L outside the box R , one has that F s,t = L outside B ∪ R ; • for every ( x, y ) ∈ B with (cid:107) ( x, y ) (cid:107) < η , we have that F s,t ( x, y ) = ( µ s x, λy ); • we assume that W uN ( p, F s,t ) = W uN ( p, F t ) for every s ∈ R , where N >
Remark 2.2.3.
The last item above is possible since F t admits an interval I ⊆W uloc ( p, F t ) such that F mt ( I ) ⊇ W uloc ( p, F t ) for some m , W uloc ( p, F s,t ) coincide withthe local strong unstable W uloc ( p, F t ) which is F uloc ( p ) and B can be chosen smallenough such that F kt ( I ) does not hit on B for ≤ k ≤ m − . Let Q = [0 , η (cid:48) ] × [0 , η (cid:48) ] be the box in ( v c , v u )-coordinates with 0 < η (cid:48) < η contained in B , where B was stablished above. Note that [0 , η ] × { } is containedin W sloc ( p, F s,t ) and { }× [0 , η (cid:48) ] is contained in W uloc ( p, F s,t ). Then, take two disjointintervals J and J as folllows, J ⊆ { } × [0 , η (cid:48) ] with p ∈ ∂J and J ⊆ { } × [0 , η (cid:48) ]on which F ms,t ( J ) ⊇ { } × [0 , η (cid:48) ] (it is possible that exist a cycle). Moreover, weassume that F s,t ( J ) contains { } × [0 , η (cid:48) ]. In particular, it contains the interval J .Let R and R be two stripe in Q of the form R = [0 , η (cid:48) ] × J and R = [0 , η (cid:48) ] × J ,see Figure 5; and if necessary we take λ large enough to guarantee that F ks,t ( R )does hit on B until the m -iterate.For simplicity, denote by F u ( x ) the strong unstable leaf containing x in Q andby F ui ( x ) the connected component in R i for i = 1 ,
2. Note that, by construction, F m + ks,t preserves the unstable foliation F ui and if t (cid:48) > t then F ms,t (cid:48) ( J ) is of the form F u ( x ) in Q with 0 < x < η (cid:48) , recalling that m was stablished in Remark 2.2.3.Hence, F m + ks,t induces a one-dimensional map f i : J i / ∼ → J i / ∼ , where x ∼ y ifand only if y ∈ F u ( x ) (it is easy to see that J i / ∼ = [0 , η (cid:48) ]), defining f i ( x ) as theleaf F m + ks,t ( F ui ( x )). Moreover, we have that | f (cid:48) | = | µ s | m + k and | f (cid:48) | = | λ | m | µ s | k .Then, we choose k ≥ t > s > | f (cid:48) | , | f (cid:48) | <
1, and f , f are as inFigure 2. Denote such F s,t simply by F which has a blender for F m + k . (cid:3) Figure 5.
The blender for an iterated of F .From now on, denote by U F the neighborhood of F where ( Q , G ) is a blender forevery G ∈ U F . Observe that the endomorphism F : T → T is a DA-endomorphismfrom a C -perturbation of L such that it coincides with the linear endomorphism L away from the support. It is easy to check that by construction one can assumethat every endomorphism G ∈ U F has the hyperbolic structure preserving the conefield C uα and verifying min (cid:107) DG ( v ) (cid:107) ≥ λ (cid:48) > Step II: crossing the blender.
Here, we guarantee that every u -arc γ admitsan iterate by G n which cross the blender. More precisely, Lemma 2.2.3.
There is a neighborhood U F of F such that for every G ∈ U F andevery u -arc γ , there exists a component ˆ γ of G n ( γ ) which cross the blender Q andintersect W sloc ( p g , G ) . EW CLASSES OF C -ROBUSTLY TRANSITIVE 15 The idea of the proof follows from the following steps. First, we observe that thestrong unstable foliation F u is minimal, that is, every leaf of F u is dense. Then, fixa large number K such that every segment on F u whose length is greater than K cross the blender. Moreover, we can fix α > γ oflength greater than K tangent to the cone-field C α also crosses the blender. Thus,now we may take a neighborhood U F of F such that every endomorphism G in italso preserves the cone-field C α and every u -arc γ has some iterated by G whichcrosses the blender. Since F restricted to R i has the form F ( x, y ) = ( f i ( x ) , λy ), wecan adapt Proposition 2.1.2 for this setting. Finally, we repeat the same argumentsas in the proof of Lemma 2.1.3 to conclude the lemma.Now we are able to prove the robustness of transitivity for F .2.2.4. Proof of Theorem A for the irrational case.
The proof follows from the samearguments as in the periodic case. We put artificially the critical points as in theprevious case in such way those lemmas in Steps I and II keep holding. Thus, onecan conclude the proof of Theorem A repeating the same arguments done for theperiodic case. (cid:3)
Acknowledgments
The authors are grateful to E. Pujals and R. Potrie for insightful commentsto improve this work. The first author would like to thank to UFBA and thesecond author to UFAL and ICTP for the nice enviroment and support during thepreparation of this work.
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