On the non-robustness of intermingled basins
OON THE NON-ROBUSTNESS OF INTERMINGLED BASINS
RA ´UL URES AND CARLOS H. V ´ASQUEZ
Abstract.
It is well-known that it is possible to construct a partially hyperbolic diffeo-morphism on the 3-torus in a similar way than in Kan’s example. It has two hyperbolicphysical measures with intermingled basins supported on two embedded tori with Anosovdynamics. A natural question is how robust is the intermingled basins phenomenon fordiffeomorphisms defined on boundaryless manifolds? In this work we study partiallyhyperbolic diffeomorphisms on the 3-torus and show that the intermingled basins phe-nomenon is not robust. Introduction
Attractors play a key role in the study of non-conservative dynamics. The description ofattractors and the properties of their basins help predict the future behaviour of the orbitsof a system. In this work we deal with physical measures i.e. an ergodic measure µ isphysical if its basin of attraction has positive volume (see Section 2 for precise definitions).We will think these measures as the attractors of our systems.In many cases, basins are (essentially) open sets and it is clear that if a point belongs tocertain regions its trajectory goes, almost surely, to an attractor that is well determined.For instance, uniformly hyperbolic diffeomorphisms exhibit a finite number of physicalmeasures and the union of their basins cover Lebesgue almost every point the ambientmanifold. Moreover, each one of their basins is an open set (modulo a set of null volume)and then, we can clearly distinguish one attractor from the others.Outside the uniformly hyperbolic world, this kind of behaviour of the basins of attractorsis no longer true. Open sets of diffeomorphisms of manifolds with boundary may haveattractors with intermingled basins. More specifically, two or more basins are dense in thesame open set. It was I. Kan [17] (See also [3] for a description of the example in terms of thepartial hyperbolicty and Lyapunov exponents) who showed for the first time the existenceof examples of partially hyperbolic endomorphisms defined on a surface and exhibiting twohyperbolic physical measures whose basins are intermingled. Moreover, he showed that Date : May 7, 2019.2000
Mathematics Subject Classification.
Primary: 37H15, 37D25, 37D30.
Key words and phrases. partial hyperbolicity, physical measures, intermigled basins, Kan example.CV was supported by the Center of Dynamical Systems and Related Fields c´odigo ACT1103 PIA -Conicyt and Proyecto Fondecyt 1130547. a r X i v : . [ m a t h . D S ] M a r R. URES AND C. H. V ´ASQUEZ such phenomenon is robust among the maps preserving the boundary. We refer the readerto [16] for a rigorous proof of Kan example and [4] for a generalization of the Kan exampleand its relation with the sign of the Schwarzian derivative. In [18] the authors shown thatthe set of points that are not attracted by either of the components in the Kan’s examplehas Hausdorff dimension less than the dimension of the phase space itself. Following thesame type of arguments, it is possible to construct a partially hyperbolic diffeomorphismdefined on a 3-manifold with boundary exhibiting two intermingled physical measures, andsuch phenomenon still can be made robust. Furthermore, it is well known that it is possibleto extend such example to the 3-torus, but in this case it is no longer robust. We describethese examples in Section 3.The existence of these examples rise the question of how robust are the intermingledbasins phenomenon for diffeomorphisms defined on boundaryless manifolds. In this work weshow that partially hyperbolic diffeomorphisms on the 3-torus having hyperbolic physicalmeasures with intermingled basins are not robust.In a recent work, Okunev [22], studied attractors in the sense of Milnor in the mostrestrictive case of C r partially hyperbolic skew products on T with an Anosov dffeomor-phisms acting on the base T . The author obtains results with the same flavour as ourswithout any explicit hypotheses about Lyapunov exponent in the central direction.We are interested in diffeomorphisms defined on a 3-dimensional manifold M , in partic-ular we put our focus on M = T . We give some basic definitions necessary to formulatethe results, but the reader can find the precise definitions, properties and more detailedinformation in Section 2 and the references therein.A diffeomorphism f : M → M is partially hyperbolic if the tangent bundle splits intothree non trivial sub-bundles T M = E uu ⊕ E c ⊕ E ss such that the strong stable sub-bundle E ss is uniformly contracted, the strong unstablesub-bundle E uu is uniformly expanded and the center sub-bundle E c may contract orexpand, but this contractions or expansions are weaker than the strong expansions andcontractions of the corresponding strong sub-bundles.It is known that there are unique foliations W uu and W ss tangent to E uu and E ss respectively [6, 15] but in general, E c , E cu = E c ⊕ E uu , and E cs = E c ⊕ E ss do not integrateto foliations (see [31]). The system is said to be dynamically coherent if there exist invariantfoliations W cu and W cs tangent to E cu and E cs respectively. Of course, if this is the case,there exists an invariant foliation tangent to E c obtained just by intersecting W cu and W cs . We will study dynamically coherent diffeomorphism with compact center leaves. Aswe mentioned above these diffeomorphisms are not always dynamically coherent althoughthere are some results providing this property. Just to mention one result, Brin, Burago, N THE NON-ROBUSTNESS OF INTERMINGLED BASINS 3 and Ivanov have shown that every absolute partially hyperbolic system (see Subsection 2.1for the definition) on the 3-torus is dynamically coherent [5].A set K ⊆ M is u -saturated if it is the union of complete strong unstable leaves. Thediffeomorphism f is accessible if every pair of points x, y ∈ M can be joined by an arcconsisting of finitely many segments contained in the leaves of the strong stable and strongunstable foliations. Assuming that the center bundle is one-dimensional, K. Burns, F. R.Hertz, J. R. Hertz, A. Talitskaya and R. Ures [8] proved that the accessibility property isopen and dense among the C r -partially hyperbolic diffeomorphisms (see also [21]) . Ourmain theorem is the following. Theorem A.
Let f ∈ Diff r ( T ) , r ≥ , be partially hyperbolic, dynamically coherentwith compact center leaves. Let µ be a physical measure with negative center Lyapunovexponent. Assume that K ⊆ T is a compact, f -invariant and u -saturated subset such that K ⊆ B ( µ ) \ supp µ . Then, K contains a finite union of periodic 2-dimensional C -tori,tangent to E u ⊕ E s . In particular f is not accessible. We say that two physical measures µ and ν with disjoint supports have intermingledbasins [17] if for an open set U ⊆ M we have Leb( V ∩ B ( µ )) > V ∩ B ( ν )) > V ⊂ U . Corollary B.
The set of dynamically coherent partially hyperbolic C r -diffeomorphisms de-fined on T , r ≥ , exhibiting intermingled hyperbolic physical measures has empty interior.Moreover, if f : T → T is isotopic to a hyperbolic automorphism, there do not existhyperbolic physical measures with intermingled basins. Closely related, Hammerlindl and Potrie [14] showed that partially hyperbolic diffeo-morphisms on 3-nilmanifold admit a unique u -saturated minimal subset. Then, f has aunique hyperbolic physical measure (see Section 2.2 for more details) and thus, it is notpossible to have the intermingled basins phenomenon. We have as corollary of their work: Corollary C. If M is a -nilmanifold , then there does not exist hyperbolic physical mea-sures with intermingled basins. This paper is organized as follows. Section 2 is devoted to introduce the main tools in theproof: partial hyperbolic diffeomorphisms, physical measures, u -measures and Lyapunovexponents. A toy example as well as Kan-like examples are revisited in Section 3. Proofsof Theorem A and Corollary B are developed in Section 4.2. Preliminaries
Partial hyperbolicity.
Throughout this paper we shall work with a partially hy-perbolic diffeomorphism f , that is, a diffeomorphism admitting a nontrivial T f -invariant
R. URES AND C. H. V ´ASQUEZ splitting of the tangent bundle
T M = E ss ⊕ E c ⊕ E uu , such that all unit vectors v σ ∈ E σx ( σ = ss, c, uu ) with x ∈ M satisfy: (cid:107) T x f v ss (cid:107) < (cid:107) T x f v c (cid:107) < (cid:107) T x f v uu (cid:107) for some suitable Riemannian metric. f also must satisfy that (cid:107) T f | E ss (cid:107) < (cid:107) T f − | E uu (cid:107) <
1. We also want to introduce a stronger type of partial hyperbolicity. We will say that f is absolutely partially hyperbolic if it is partially hyperbolic and (cid:107) T x f v ss (cid:107) < (cid:107) T y f v c (cid:107) < (cid:107) T z f v uu (cid:107) for all x, y, z ∈ M .For partially hyperbolic diffeomorphisms, it is a well-known fact that there are foliations W σ tangent to the distributions E σ for σ = ss, uu . The leaf of W σ containing x will becalled W σ ( x ), for σ = ss, uu .In general it is not true that there is a foliation tangent to E c . Sometimes there is nofoliation tangent to E c . Indeed, there may be no foliation tangent to E c even if dim E c = 1(see [31]). We shall say that f is dynamically coherent if there exist invariant foliations W cσ tangent to E cσ = E c ⊕ E σ for σ = ss, uu . Note that by taking the intersection ofthese foliations we obtain an invariant foliation W c tangent to E c that subfoliates W cσ for σ = s, u . In this paper all partially hyperbolic diffeomorphisms will be dynamicallycoherent.We shall say that a set X is σ -saturated if it is a union of leaves of the strong foliations W σ for σ = ss or uu . We also say that X is su -saturated if it is both s - and u -saturated.The accessibility class of the point x ∈ M is the minimal su -saturated set containing x .In case there is some x ∈ M whose accessibility class is M , then the diffeomorphism f issaid to have the accessibility property . This is equivalent to say that any two points of M can be joined by a path which is piecewise tangent to E ss or to E uu .2.2. Physical measures, u -measures, Lyapunov exponents. In this section we con-sider f : M → M be a diffeomorphism, not necessarily partially hyperbolic, defined on theriemannian manifold M . We denote by Leb the normalized volume form on M .A point z ∈ M is Birkhoff regular if the Birkhoff averages(2.1) ϕ − ( z ) = lim n →∞ n n − (cid:88) k =0 ϕ ( f − k ( z )) , (2.2) ϕ + ( z ) = lim n →∞ n n − (cid:88) k =0 ϕ ( f k ( z ));are defined and ϕ − ( z ) = ϕ + ( z ) for every ϕ : M → R continuous. We denote by R ( f ) theset of Birkhoff regular points of f . Birkhoff Ergodic Theorem [19, 33], implies that the set N THE NON-ROBUSTNESS OF INTERMINGLED BASINS 5 R ( f ) has full measure with respect to any f -invariant measure ξ . When ξ is an ergodicmeasure, ϕ − ( z ) = ϕ + ( z ) = (cid:90) M ϕ dξ, for every z in a ξ -full measure set R ( ξ ).If ξ is an f -invariant measure, the basin of ξ is the set B ( ξ ) = { z ∈ M : lim n →∞ n n − (cid:88) k =0 ϕ ( f k ( z )) = (cid:90) M ϕ dξ, for all ϕ ∈ C ( M, R ) } If ξ is an f -invariant ergodic measure, then R ( ξ ) ⊆ B ( ξ ),and so B ( ξ ) has full ξ -measure.An f -invariant probability measure µ is physical if its basin B ( µ ) has positive Lebesguemeasure on M [3, 34]. A physical measure is said to be hyperbolic if all its Lyapunovexponents are nonzero [2]. In the setting of partially hyperbolic diffeomorphims defined ona 3-dimensional manifold, a physical measure is hyperbolic if λ c ( µ ) = (cid:90) log (cid:107) Df | E c (cid:107) dµ (cid:54) = 0 . A point x ∈ M is Lyapunov regular if there exist an integer p ( x ) ≤ dim M , numbers χ ( x ) < · · · < χ p ( x ) ( x ) , and a decomposition(2.3) T x M = p ( x ) (cid:77) i =1 H i ( x )into subspaces H i ( x ) such that Df ( x ) H j ( x ) = H j ( f ( x )), and for every v ∈ H j ( x ) \ { } (2.4) χ j ( x ) = lim n →±∞ n log (cid:107) Df n ( x ) v (cid:107) . Denote by Λ( f ) the set of Lyapunov regular points. The numbers χ ( ξ ) < · · · < χ p ( x ) ( x ) , are called the Lyapunov exponents of x . The splitting (2.3) is called Oseledets decomposition and the subspaces H i ( x ) are called Oseledets subespaces at x . Oseledet’s Theorem [23, 19]guarantee that the set Λ( f ) has full measure with respect any invariant measure. In generalthe functions x → χ j ( x ), x → H j ( x ), x → p ( x ) and x → dim H j ( x ) are measurable.Nevertheless, if ξ is an ergodic invariant measure for f , there is a subset Λ( ξ ) ⊆ Λ( f ),such that ξ (Λ( ξ )) = 1 and there exist an integer p ( ξ ) ≤ dim M , subspaces H ( ξ ) , ...H p ( ξ ),numbers χ ( ξ ) < · · · < χ p ( ξ ) such that for every x ∈ Λ( ξ ), we have • p ( x ) = p ( ξ ); • dim H j ( x ) = dim H j ( ξ ), for every j = 1 , . . . , p ( ξ ); • χ j ( x ) = χ j ( ξ ), for every j = 1 , . . . , p ( ξ ); R. URES AND C. H. V ´ASQUEZ
An ergodic measure ξ is hyperbolic if χ j ( ξ ) (cid:54) = 0, j = 1 , . . . , p ( ξ ). In such case, for each x ∈ Λ( ξ ) we set H s ( x ) = (cid:77) χ j ( ξ ) < H j ( x ) , and H u ( x ) = (cid:77) χ j ( ξ ) > H j ( x ) . We have dim H s ( x ) = s ( ξ ), dim H u ( x ) = u ( ξ ) are constant and s ( ξ ) + u ( ξ ) = dim M . Thefunction x → H s ( x ) and x → H u ( x ) are measurables. If f is C r , r >
1, Pesin’s Theory[10, 24, 25, 29] guarantee the existence of invariant sub-manifolds W s ( x ), W u ( x ) tangent to H s ( x ) and H u ( x ) respectively. More precisely, for every x ∈ Λ( ξ ) there is a C r embeddeddisk W s loc ( x ) through x such that • W s loc ( x ) is tangent to H s ( x ) at x , • f ( W s loc ( x )) ⊆ W s loc ( f ( x )), • The stable set W s ( x ) = ∪ ∞ n =0 f − n ( W s loc ( f n ( x ))) . • There exist constant C ( x ) > τ ( x ) such that, for every x , x ∈ W s loc ( x )(2.5) dist( f k ( x ) , f k ( x )) ≤ C ( x ) e − kτ ( x ) dist( x , x ) . The C r disk W s loc ( x ) is called Pesin stable manifold . Similarly, every x ∈ Λ( ξ ) has an Pesin unstable manifold W u loc ( x ) satisfying the corresponding properties with f − in placeof f .The Pesin manifolds above may be arbitrarily small, and they vary measurably on x .For any integer n ≥
1, we may find hyperbolic blocks Λ n ( ξ ) ⊆ Λ( ξ ) such that • Λ n ( ξ ) ⊆ Λ n +1 ( ξ ), • ξ (Λ n ( ξ )) →
1, as n → ∞ . • The the size of the embedded disk W s loc ( x ) is uniformly bounded from zero for each x ∈ Λ n ( ξ ). Moreover, for every x ∈ Λ n ( ξ ), C ( x ) < n and τ ( x ) > /n in (2.5).Analogous properties are satisfied by the unstable Pesin’s manifold W u loc ( x ). • The disk W s loc ( x ) and W u loc ( x ) vary continuously with x ∈ Λ n ( ξ ).Most important, the holonomy maps associated to the Pesin stable lamination W sP = { W s loc ( x ) } are absolutely continuous. More precisely, fix an integer n ≥
1, a hyperbolicblock Λ n ( ξ ) and a point x ∈ Λ n ( ξ ). For x , x ∈ W s loc ( x ) close to x , let Σ and Σ be smallsmooth discs transverse to W s loc ( x ) at x and x respectively. The holonomy map π s : ˜Σ ⊆ Σ → Σ defined on the points y ∈ ˜Σ = Σ ∩ Λ n ( ξ ) by associate π s ( y ), the unique point inΣ ∩ W s ( y ). If f is C r , r >
1, then every holonomy map π s as before is absolutelycontinuous [24, 29]. Of course, a dual statement holds for the unstable lamination. N THE NON-ROBUSTNESS OF INTERMINGLED BASINS 7
In our setting, f is a C r -partially hyperbolic diffeomorphism, r >
1, with splitting
T M = E ss ⊕ E c ⊕ E uu , where dim E σ = 1, σ = ss, c, uu . Let ξ be an ergodic measure andwe consider any point x ∈ Λ( ξ ). Then p ( x ) = p ( ξ ) = 3 and H ( x ) = E s ( x ), H ( x ) = E c ( x )and H ( x ) = E uu ( x ). Moreover χ ( ξ ) =: λ s , χ ( ξ ) =: λ u and χ ( x ) = lim n →±∞ n log (cid:107) Df n ( x ) | E cx (cid:107) =: λ c ( x ) , is called the center Lyapunov exponent at x . If we take x ∈ R ( ξ ) ∩ Λ( ξ ), since dim E c = 1we obtain that(2.6) χ ( ξ ) = λ c ( ξ ) := λ c ( x ) = (cid:90) log (cid:107) Df | E c (cid:107) dξ. If we assume λ c ( ξ ) <
0, then H s ( x ) = E ss ( x ) ⊕ E c ( x ) and H u ( x ) = E uu ( x ) for every x ∈ Λ( ξ ). The local strong stable manifold W ss loc ( x ) is an embedded curve inside the Pesinstable manifold W s loc ( x ) which is a surface. On the other hand, the Pesin unstable manifold W u loc ( x ) coincides with the strong unstable manifold W uu loc ( x ), for every x ∈ Λ( ξ ). Of course,analogous statement holds if we assume λ c ( ξ ) > f is partially hyperbolic and dim E uu ≥
1. An f -invariant probabilitymeasure µ is a u -measure if the conditional measures of µ with respect to the partitioninto local strong-unstable manifolds are absolutely continuous with respect to the Lebesguemeasure along the corresponding local strong-unstable manifold. If f is a C r partiallyhyperbolic diffemorphism, r ≥
2, then there exist u -measures for f [26]. Several propertiesof the u -measures are well know (see for instance [3], Section 11.2.3 and the referencestherein, for a detailed presentation of such properties). For instance, the support of any u -measure is a u -saturated, f -invariant, compact set. If µ is a u -measure, then its ergodiccomponents are u -measures as well. Furthermore, the set of u -measures for f is a compact,convex subset of the invariant measures. Moreover, every physical measure for f must bea u -measure.It is well know that if µ is an ergodic u -measure with negative center Lyapunov exponent,then, µ is a physical measure [34]. Conversely, if µ is a physical measure with negativecenter Lyapunov exponent, then µ is an ergodic u -measure.3. Examples
In this section we show some examples that motivated this paper. In the first example(Anosov times Morse-Smale) there are no intermingled basins but there is a u -saturatedset in the boundary of one of them. Of course, we know a priori that this set consists oftori and it is not difficult to show that this situation is not robust. This example jointlywith Kan’s was a source of inspiration to obtain Theorem A. This is the easiest case wherethe theorem works. Observe that there is only one physical measure. In the second case(Kan-like example) the basins are intermingled. R. URES AND C. H. V ´ASQUEZ
Toy Example.
In the 3-torus T × S , we consider the C r -diffeomorphism, r ≥ F : T → T defined by F ( x, t ) = ( Ax, ξ ( t )) , where A : T → T is a linear Anosov diffeomorphism with eigenvalues | λ sA | < < | λ uA | ,and ξ : S → S is a Morse-Smale diffeomorphisms with having exactly two hyperbolic fixedpoints, a source p ∈ S and a sink q ∈ S , satisfying ∂W u ( p, ξ ) = { q } and ∂W s ( q, ξ ) = { p } .We assume that F satisfies: | λ sA | < | Dξ ( t ) | < | λ uA | , for every t ∈ S . That means, F is a partially hyperbolic diffeomorphism exhibiting a center foliation bycompact leaves (circles). Furthermore, F has a foliation by smooth 2-tori tangent to the E s ⊕ E u -sub-bundle. In particular, one of such leaves, the torus T × { q } , is the onlyattractor of F . The dynamics restricted to T × { q } is hyperbolic, in fact, is given by A . Then, it supports the unique hyperbolic u -measure µ q for F (actually the Lebesguemeasure on T × { q } ) having negative center Lyapunov exponent and so, it is physical. If B A ( µ ) denotes the basin of µ in the 2-torus T × { q } under the Anosov dynamics givenby A then, the basin of µ q in T is B ( µ q ) = B A ( µ q ) × { S \ { p }} , which is an open set modulus a set of zero Lebesgue measure in T . The boundary of B ( µ q )contains the invariant 2-torus T × { p } which is the only hyperbolic repeller of F . Thisinvariant torus is also a u -saturated set, tangent to E s ⊕ E u . The dynamics restricted to T × { p } is again hyperbolic and then, it supports a u -measure µ p for F (actually Lebesguemeasure on T ) but it is not physical.Theorem A prevents the existence of such a u -saturated set from being robust. After atypical C -perturbation, the new map G is partially hyperbolic and dynamically coherent.In fact, G has a center foliation by compact leaves by classical results of normally hyperbolicfoliations [15].Typically G does not preserve the invariant foliation by 2-tori tangent to E s ⊕ E u , whichexists for F . Nevertheless G has two invariant compact subset Λ p and Λ q , the respectivecontinuations of the hyperbolic basic sets T × { p } and T × { q } . Of course, the dynamicsof F | T × { p } and G | Λ p are C -conjugated, so Λ p is (homeomorphic to) a continuous torus,and the dynamics of G in Λ p is uniformly hyperbolic. The set Λ p remains to be a hyperbolicrepeller and so s -saturated, but in general it is not u -saturated.Similar conclusions hold for Λ q , the hyperbolic attractor of G . It is a topological 2-torus, u -saturated, and it supports the unique physical measure of G . Note that the topologicaltorus Λ p is contained in the boundary of the basin B ( µ Gq ), but, in general, Λ p is no longera u -saturated set. N THE NON-ROBUSTNESS OF INTERMINGLED BASINS 9
Kan-like Examples.
In [17] Kan provided the first examples of partially hyperbolicmaps with intermingled basin. In this section we present the Kan’s examples with somevariations, following [3], Section 11.1.1.3.2.1.
Kan’s example: Endomorphism.
The Kan’s example corresponds to a partially hy-perbolic endomorphism defined on a surface with boundary exhibiting two intermingledhyperbolic physical measures. Consider the cylinder M = S × [0 , K : M → M themap defined by K ( θ, t ) = ( kθ (mod Z ) , ϕ ( θ, t )) , where k ≥ p, q ∈ S are two different fixed points of θ → kθ (mod Z ) and ϕ : M → [0 ,
1] is C r , r ≥
2, satisfying the following conditions:[K1] For every θ ∈ S we have ϕ ( θ,
0) = 0 and ϕ ( θ,
1) = 1.[K2] The map ϕ ( p, · ) : [0 , → [0 ,
1] has exactly two fixed points, a hyperbolic source at t = 1 and a hyperbolic sink in t = 0. Analogously, the map ϕ ( q, · ) : [0 , → [0 , t = 1 and a hyperbolic source in t = 0.[K3] For every ( θ, t ) ∈ M , | ∂ t ϕ ( θ, t )) | < k , and[K4] (cid:90) log | ∂ t ϕ ( θ, | dθ < (cid:90) log | ∂ t ϕ ( θ, | dθ < θ -direction is given by θ → kθ (mod Z ), so it is uniformly expanding.From [K3] we conclude that the map K is partially hyperbolic: The derivative in the t -direction is dominated by the derivative in the θ -direction. Condition [K1] means K preserves the boundary. Then, each one of the boundary circles S × { } and S × { } supports an absolutely continuous invariant probability measure µ and µ , respectively.Condition [K4] implies that µ and µ have negative Lyapunov exponent in the t -direction.So they are physical measures. Moreover, their basin are intermingled. Magic comesfrom condition [K2]: Take any curve γ inside the open cylinder and transverse to the t -direction. We can assume, up to taking some forward iterates, that γ crosses (transversally)the segments W s ( p,
0) = { p } × [0 ,
1) and W s ( q,
0) = { q } × (0 , f is uniformly expanding along the θ direction and the angle between γ and the t -directiongoes to π due to the domination. Then, there is a forward iterate of γ that intersects thebasin of µ , in a set of positive Lebesgue measure (in γ ), because γ intersects transversally W s ( p,
0) = W ss ( p ) × [0 , γ also intersects transversally W s ( q,
0) = W ss ( q ) × (0 , γ intersects the basin of µ in a set of positive Lebesgue measure (See Figure 1).Fubini’s theorem completes the argument.This example is robust among the maps defined on the cylinder preserving the bound-aries. Indeed, for r ≥
1, any map ˜ K : M → M , C r close to K and preserving theboundaries can be written as ˜ K ( θ, t ) = ( F ( θ, t ) , ψ ( θ, t )) , Figure 1.
Kan example in the cylinder S × [0 , F : M → S is expanding along the θ -direction and ψ : M → [0 ,
1] preservs theboundaries, that means ψ satisfies [K1]. Moreover, if ψ is chosen C r close enough of ϕ ,then also their derivatives ∂ t ψ ( θ, t ) and ∂ t ϕ ( θ, t ) are close for every ( θ, t ) ∈ M and so ψ satisfies [K3] and [K4] above. The two different fixed points of θ → kθ (mod Z ), p, q ∈ S ,have continuations ˜ p, ˜ q ∈ S and the map ψ (˜ p, · ) : [0 , → [0 ,
1] has exactly two fixedpoints, a hyperbolic source at t = 1 and a hyperbolic sink in t = 0. Analogously, themap ψ (˜ q, · ) : [0 , → [0 ,
1] has exactly two fixed points, a hyperbolic sink at t = 1 and ahyperbolic source in t = 0. Then, ψ satisfies [K2]. Arguing as before, we conclude that ˜ K exhibits two intermingled hyperbolic physical measures supported on the boundary.3.2.2. Kan’s example: Diffeomorphisms on a manifold with boundary.
The next example,corresponds to a partially hyperbolic diffeomorphism defined on a 3-manifold with bound-ary exhibiting two intermingled physical measures. The idea is to adapt the previousexample, replacing S with the torus T and the expanding map θ → kθ (mod Z ) with ahyperbolic automorphism of the 2-torus having at least two fixed points. More precisely,we can consider N = T × [0 ,
1] and diffeomorphisms K D ( z, t ) = ( Az, ψ ( θ, t )) , where A : T → T is a hyperbolic automorphism, and ψ : N → [0 ,
1] is C r , r ≥ N THE NON-ROBUSTNESS OF INTERMINGLED BASINS 11 [KD1] For every z ∈ T we have ψ ( z,
0) = 0 and ψ ( z,
1) = 1.[KD2] For p, q ∈ T , fixed points of A , we assume that the map ψ ( p, · ) : [0 , → [0 , t = 1 and a sink in t = 0. Analogously,the map ψ ( q, · ) : [0 , → [0 ,
1] has exactly two fixed points, a sink at t = 1 and asource in t = 0.[KD3] For every ( z, t ) ∈ M , (cid:107) A − (cid:107) − < | ∂ t ψ ( x, t )) | < (cid:107) A (cid:107) , and[KD4] (cid:90) T log | ∂ t ψ ( z, | dz < (cid:90) T log | ∂ t ψ ( z, | dx < z -direction of K D is uniformly hyperbolic. From [KD3]we conclude that the map K D is partially hyperbolic: The derivative in the t -direction isdominated by the derivative in the unstable direction of A and the stable direction of A isdominated by the derivative in the t -direction. Condition [KD1] means K D preserves eachboundary torus. Then both boundary torus T × { } and T × { } support the measures µ and µ corresponding to the Lebesgue measure in the torus. Condition [KD4] implies that µ and µ have negative Lyapunov exponent in the center direction. So they are physicalmeasures.As before, their basins are intermingled. The argument is the same: Take any curve γ in the interior of N and transverse to the E cs distribution. Up to some forward iterates, γ crosses (transversally) the surfaces W sloc ( p,
0) = W ssloc ( p ) × [0 ,
1) and W sloc ( q,
0) = W ssloc ( q ) × (0 , f is uniformly expanding along the unstable direction and thedomination improves the angle between γ and the center-stable direction. Then, there is aforward iterate of γ that intersects the basin of µ in a set of positive Lebesgue measure (in γ ), because γ intersects transversally the stable manifold W sloc ( p, γ also intersectstransversally the stable manifold W s ( q, γ intersects the basin of µ in a set ofpositive Lebesgue measure. Fubini’s theorem complete the argument.As before, this example is robust among the diffeomorphisms defined on N preservingthe boundary tori.3.2.3. Kan-like example: Diffeomorphisms on a boundaryless manifold.
The same con-struction can be done if N is replaced with T = T × S (or even the mapping torus of ahyperbolic diffeomorphism) and ψ : N → [0 ,
1] is replaced with ϕ : T × S → S . Then,the four conditions are:[KB1] For every z ∈ T we have ϕ ( z,
0) = 0 and ϕ ( z, ) = .[KB2] For p, q ∈ T , fixed point of A , we assume that the map ϕ ( p, · ) : S → S hasexactly two fixed points, a source at t = and a sink in t = 0. Analogously, themap ϕ ( q, · ) : S → S has exactly two fixed points, a sink at t = and a source in t = 0.[KB3] For every ( z, t ) ∈ M , (cid:107) A − (cid:107) − < | ∂ t ϕ ( x, t )) | < (cid:107) A (cid:107) , and [KB4] (cid:90) T log | ∂ t ϕ ( z, | dz < (cid:90) T log | ∂ t ϕ ( z,
12 ) | dx < T with exactly k ≥ µ , . . . , µ k whose basinsare all intermingled (and dense on the whole torus), in fact, for every open set A ⊆ T andevery i (cid:54) = j ∈ { , . . . , k } Leb( A ∩ B ( µ i )) > A ∩ B ( µ j )) > . Their example is partially hyperbolic in the following broad sense: the tangent spacehas an invariant splitting T T = E cs ⊕ E u where E u dominates E cs but the sub-bundle E cs is indecomposable into dominated sub-bundles.We remark that partially hyperbolic diffeomorphisms on surfaces do not admit inter-mingled hyperbolic physical measures [30]. The situation is different in the absence ofdomination as showed by Fayad [11]. Inspired in the Fayad example, Melbourne andWindsor [20] give a family of C ∞ -diffeomorphisms on T × S with arbitrary number ofphysical measures with intermingled basins.Motivated by the latter situation, we say that a partially hyperbolic diffeomorphism f isa Kan-like differmorphisms if there exist, at least, two hyperbolic physical measures withintermingled basins.4.
Proof of Theorem A and Corollary B
Let f ∈ Diff r ( M ), r ≥
2, be partially hyperbolic and dynamically coherent with compactcenter leaves. Let µ be a hyperbolic physical measure for f with λ c ( µ ) <
0. For furtheruse let Λ = ∪ n Λ n where Λ n are Pesin blocks and µ (Λ) = 1. We assume that Λ is invariantand its points are regular both in the sense of Pesin’s Theory as in the sense of Birkhoff’sTheorem. Moreover, we will assume that every x ∈ Λ n is a Lebesgue density point of W u ( x ) ∩ Λ n .For E ⊆ M measurable, W s ( E ) denotes the union of Pesin’s stable manifolds W s ( x ) ofpoints x ∈ E . Observe that W s ( E ) is invariant if E is invariant.First, for the sake of completeness, we will prove the following lemma. We thank thereferee for provide us the argument of the proof. Lemma 4.1. B ( µ ) ⊂ W s (Λ) .Proof. Suppose x ∈ B ( µ ). Fix m ≥
1. Then, it is not difficult to see there is a sequence n k such that the distance between f n k ( x ) and Λ m converges to 0. Indeed, if there is δ > f n ( x ) to Λ m is greater than 0 you can construct a continuous N THE NON-ROBUSTNESS OF INTERMINGLED BASINS 13 which that takes the value 1 for every point in Λ m and 0 if the distance to Λ m is greateror equal to δ . Since Λ m has positive µ -measure this contradicts the fact that x ∈ B ( µ ).As Pesin stable manifolds are of uniform size for points in Λ m , there is y k ∈ W uloc ( f n k ) ∩ W s (Λ) for any k large enough. Clearly f − n k ( y k ) converges to x proving the lemma. (cid:3) Denote by M c the space of center curves, that is, the quotient space obtained by therelation of equivalence y ∼ x if they are in the same center manifold. We denote by X thespace of compact subsets of M . Given a u -saturated closed subset K ⊆ M , we define thefunction Φ K : M c → X by Φ K (¯ x ) = K ∩ ¯ x . Observe that this intersection is nonempty forevery ¯ x ∈ M c .Since K is closed we have that Φ K is an upper semicontinuous function. This impliesthat Φ K has a residual set of points of continuity.On the other hand, since K is saturated by strong unstable leaves and the unstableholonomy is continuous, the set of continuity points of Φ K is also saturated by strongunstable leaves. More precisely, if ¯ x is a point of continuity of Φ K , then for every y ∈ W u (¯ x )we have that ¯ y ∈ M c is also a point of continuity of Φ K . Lemma 4.2.
For every x ∈ W s (Λ) , there is a center arc [ x, y ] c ⊆ W s ( y ) with y ∈ supp( µ ) .Proof. Let x ∈ W s (Λ m ). Taking iterates for the future, and recalling that almost everypoint returns infinitely many times to a positive measure set, we can assume that x ∈ W sε ( y (cid:48) ) with y (cid:48) ∈ Λ m where ε is the uniform size of the Pesin stable manifolds of the pointsof the block Λ m . Close to y (cid:48) we take z ∈ W s ( y (cid:48) ) ∩ supp ( µ ), with dist( y (cid:48) , z ) < ε , and suchthat ¯ z = W c ( z ) is a continuity point of Φ supp ( µ ) . In particular, there is a δ > z, w ) < δ then, there exists p ∈ W c ( w ) ∩ supp ( µ ) with dist( p, z ) < ε .Let H = Λ m ∩ W uu ε ( y (cid:48) ) and G = W s(cid:15) ( H ) ∩ B δ/ ( z ) ∩ supp ( µ ). The absolute continuityof the partition by Pesin’ stable manifolds implies that µ ( G ) >
0. Then, the ergodicityof the measure implies that there are infinitely many iterates of y (cid:48) that belong to G . Inparticular, there is an n such that f n ( y (cid:48) ) ∈ G and dist( f n ( x ) , f n ( y (cid:48) )) < δ/
2. Thus, weobtain that f n ( x ) ∈ B δ ( z ) ∩ W sε ( H ). The fact that f n ( x ) ∈ B δ ( z ) implies that there is v ∈ W c ( f n ( x )) ∩ supp µ , such that dist( f n ( x ) , v ) < ε . Since f n ( x ) ∈ W sε ( H ) we have thatcorresponding center arc [ f n ( x ) , v ] c is completely contained in a Pesin stable manifold. Wetake y = f − n ( v ) and this gives the conclusion of the lemma for the points of W s (Λ). (cid:3) In what follows we consider K ⊆ T satisfying the hypotheses in Theorem A. That is, K is a compact, f -invariant and u -saturated subset such that K ⊆ B ( µ ) \ supp µ . Ourstrategy to prove Theorem A will be to study the intersections of the set K with the centermanifolds of f . Lemma 4.3.
There is a h > such that if we have three distinct points x, y, z ∈ Φ K ( ¯ w ) then at least two of them are a c -distance larger than h . Proof.
As we have already mentioned we will use that W ss ( W uu ( x )), when consideredin the universal cover, is topological surface topologically transverse to the center leaves[12, 13].Let’s begin with the proof. Suppose on the contrary that for every h there are w andthree points x, y, z ∈ Φ K ( ¯ w ) with dist c ( u, v ) < h for every pair of points { u, v } ⊂ { x, y, z } .Take the topological surfaces W ssloc ( W uuloc ( x )), W ssloc ( W uuloc ( y )) and W ssloc ( W uuloc ( z )). Withoutloose of generality we can assume that y is in the center arc that joins x and z and haslength less than h . Take k > K, supp µ ) > k and suppose that h (cid:28) k .Since y ∈ B ( µ ), Lemma 4.1 implies that it can be approximated by a point q belonging to W s (Λ). By Lemma 4.2 we have that q can be joined to supp ( µ ) by a center arc completelycontained in B ( µ ). Observe that q is very close to y ∈ K and then, the length of this centerarc is greater than, say, k/
2. Still much larger than h . This implies that the center arcjoining q and supp ( µ ) must intersect either W ssloc ( W uuloc ( x )) or W ssloc ( W uuloc ( z )) (See Figure 2). Figure 2.
The center arc (in red) joining q and supp ( µ ) must intersect thelocal planes.This is a contradiction because these sets are in the complement of B ( µ ), since the ω -limits of its points are in K . This ends the proof of the lemma. (cid:3) The preceding lemma has an immediate and important consequence that we state asproposition.
N THE NON-ROBUSTNESS OF INTERMINGLED BASINS 15
Proposition 4.4.
Let µ be an ergodic u -measure with negative center exponent and K aninvariant u -saturated set such that K ⊆ B ( µ ) \ supp µ . Then, the intersection of K witheach center manifold consists of finitely many points. Our next lemma says that the number of points of the intersection of a u -minimal subset J of K with each center manifold is constant. Lemma 4.5.
Let J ⊆ K be u -minimal compact set. Then J (¯ x ) does not depend on ¯ x Proof.
We want to show that the function J is constant in an open set. If this is thecase, the u -minimality of J and the u -invariance of J will imply the proposition.Observe that, a priori, the semicontinuity of Φ J does not imply directly the propositionbecause it is not enough to conclude the semicontinuity of J .Let ¯ x be a point of continuity of Φ J . Continuity at ¯ x implies that J (¯ y ) ≥ J (¯ x ) if¯ y is close enough to ¯ x . The u -minimality, again, implies the inequality for every ¯ y ∈ M c .Suppose that the function J is not constant. Then, there is a dense set D ⊆ M c suchthat for ¯ y ∈ D we have that J (¯ y ) > J (¯ x ). Continuity at ¯ x implies that there are apoint x ∈ ¯ x , a sequence ¯ y n → ¯ x and for each integer n ≥
1, a pair of points y n , y n ∈ ¯ y n ∩ J so that both sequences ( y in ), i = 1 ,
2, converge to x . Then, taking N large enough we canchoose a center curve with two points y N := y and y N := y a very small c -distance. Wewill argue in a similar way to the arguments of the proof of Lemma 4.3. We want to obtainthree points that are very close to each other in the same center manifold and surfacesthrough them that are not in B ( µ ), to arrive to a contradiction with Lemma 4.2.Since J is u -minimal we can find z ∈ W uu ( x ) very close to y . Continuity of the holonomygives that there are center manifolds converging to the center manifold of z and pairs ofpoints w n , w n of J in each of these center manifolds converging to z . Finally, fix an integer L ≥ c distance between w L and w L is much smallerthan the one between y and y ) and call w = w L and w = w L . Denote ¯ w the center leafthat contains { w , w } . Because of the choices we have made, W ssloc ( W uuloc ( y )) intersects ¯ w in a point w that is close to w and w but at a greater distance than dist c ( w , w ). Thatmeans that one of the two points w , w lies in between the other two (See figure 3).Now, arguing as in Lemma 4.3 we arrive to a contradiction. (cid:3) Proof of Theorem A.
Let J ⊆ K be u -minimal and closed. Lemma 4.5 shows that J is locally the graph of a continuous function and then, it is a closed topological surfacetopologically transverse to the center foliation. Since it is foliated by unstable leaves, thatare lines, we have that J is a torus. Moreover, Proposition 4.4 implies that the torus J isperiodic. Thus, all that remains is to prove that the strong stable manifolds of the pointsof J are completely contained in J . Figure 3. W ssloc ( W uuloc ( y )) intersects ¯ w in a point w that is close to w and w .As J ⊆ K is periodic, we can take an iterate n ≥ f n ( J ) = J . By simplicitywe assume that n = 1. Suppose that there is a point x ∈ J such that its strong stablemanifold W ss ( x ) has a point y that does not belong to J . Since J is closed, there existsan open neighbourhood V ⊆ M of y such that V ∩ J = ∅ . By the continuity of the strongstable foliation, reducing V if necessary, we can find an open neighbourhood U ⊆ M of x with the property that the strong stable manifold of every point in V has a point in U , inparticular, in J . We know that J ⊆ K ⊆ B ( µ ) \ supp µ , then V ∩ B ( µ ) (cid:54) = ∅ . Hence, thereis z ∈ V ∩ B ( µ ) and if we take ˜ z ∈ W ss ( z ) ∩ J , then ˜ z ∈ B ( µ ) (See Figure 4).In particular, ˜ z ∈ J and its omega limit is contained in supp ( µ ). Since J is f -invariant,then ∅ (cid:54) = supp ( µ ) ∩ J ⊆ K which contradicts the hypotheses J ⊆ K ⊆ B ( µ ) \ supp µ . Thisfinishes the proof of the Theorem A. (cid:3) Proof of Theorem B.
Let µ and ν be two hyperbolic physical measures. Recall that theirsupports are compact, f -invariant and u -saturated subsets.First of all, observe that neither µ nor ν can have positive center Lyapunov exponent.This is a consequence of the well-known fact that under our hypotheses the basin of attrac-tion of such a measure would be essentially open (See for instance [7] where the conservative N THE NON-ROBUSTNESS OF INTERMINGLED BASINS 17
Figure 4.
Graphic representation of the proof of Theorem A.case is discussed with details and recently [1] for a discussion about the non conservativecase.) .Suppose that the center exponents are negative. If their basins are intermingled thensupp ν ⊆ B ( µ ) \ supp µ . Indeed, it is not difficult to see that the definition of intermingledbasins implies that there is a point of the stable manifold (in the sense of Pesin) of aregular point of ν that is accumulated by points of the basin of µ . Since ν is ergodic theorbit of a regular point is dense in its support. By forward iteration we obtain the desiredinclusion. Then, as consequence of Theorem A applied to K = supp ν , f is not accessible.As mentioned above accessibility is an open an dense property, and then we obtain thefirst assertion.For the second statement, the works of A. Hammerlindl [12] and R. Potrie [27] provedthat the center foliation of every dynamically coherent partially hyperbolic diffeomorphismon the 3-torus is homeomorphic to the corresponding foliation of a linear toral automor-phism. As a consequence, there are two possibilities: either the center foliation is by circlesor the diffeomorphism is homotopic to a hyperbolic automorphism, it is always dynami-cally coherent and the center foliation is by lines. We have already studied the first case.In the second case, Potrie [28] (see also [32]) proved that if f is isotopic to a hyperbolicautomorphism, there is a unique minimal u -saturated set. This implies that f has at mostone physical measure with negative center exponent. (cid:3) Acknowledgement.
The authors would like to thank the anonymous reviewer for their help-ful and constructive comments.
References [1] M. Andersson and C. H. V´asquez. On mostly expanding diffeomorphisms.
ArXiv e-prints :1512.01046,December 2015.[2] L. Barreira and Y. Pesin.
Lyapunov exponents and smooth ergodic theory , volume 23 of
UniversityLecture Series . American Mathematical Society, Providence, RI, 2002.[3] C. Bonatti, L. D´ıaz, and M. Viana.
Dynamics beyond uniform hyperbolicity , volume 102 of
Encyclopae-dia of Mathematical Sciences . Springer-Verlag, Berlin, 2005. A global geometric and probabilisticperspective, Mathematical Physics, III.[4] A. Bonifant and J. Milnor. Schwarzian derivatives and cylinder maps. In
Holomorphic dynamics andrenormalization , volume 53 of
Fields Inst. Commun. , pages 1–21. Amer. Math. Soc., Providence, RI,2008.[5] M. Brin, D. Burago, and S. Ivanov. Dynamical coherence of partially hyperbolic diffeomorphisms ofthe 3-torus.
J. Mod. Dyn. , 3(1):1–11, 2009.[6] M. Brin and Y. Pesin. Partially hyperbolic dynamical systems.
Izv. Akad. Nauk SSSR Ser. Mat. ,38:170–212, 1974.[7] K. Burns, D. Dolgopyat, and Y. Pesin. Partial hyperbolicity, Lyapunov exponents and stable ergod-icity.
J. Statist. Phys. , 108(5-6):927–942, 2002. Dedicated to David Ruelle and Yasha Sinai on theoccasion of their 65th birthdays.[8] K. Burns, F. Rodriguez Hertz, J. Rodriguez Hertz, A. Talitskaya, and R. Ures. Density of accessibilityfor partially hyperbolic diffeomorphisms with one-dimensional center.
Discrete Contin. Dyn. Syst. ,22(1-2):75–88, 2008.[9] D. Dolgopyat, M. Viana and J. Yang. Geometric and measure-theoretical structures of maps withmostly contracting center.
Comm. Math. Phys.
Geometricdynamics (Rio de Janeiro, 1981) , volume 1007 of
Lecture Notes in Math. , pages 177–215. Springer,Berlin, 1983.[11] B. Fayad. Topologically mixing flows with pure point spectrum. In
Dynamical systems. Part II , Pubbl.Cent. Ric. Mat. Ennio Giorgi, pages 113–136. Scuola Norm. Sup., Pisa, 2003.[12] A. Hammerlindl.
Leaf conjugacies on the torus . ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)–University of Toronto (Canada).[13] A. Hammerlindl. Leaf conjugacies on the torus.
Ergodic Theory Dynam. Systems , 33(3):896–933, 2013.[14] A. Hammerlindl and R. Potrie. Pointwise partial hyperbolicity in three-dimensional nilmanifolds.
J.Lond. Math. Soc. (2) , 89(3):853–875, 2014.[15] M. Hirsch, C. Pugh, and M. Shub.
Invariant manifolds . Springer-Verlag, Berlin, 1977. Lecture Notesin Mathematics, Vol. 583.[16] Y. Ilyashenko, V. Kleptsyn, and P. Saltykov. Openness of the set of boundary preserving maps of anannulus with intermingled attracting basins.
J. Fixed Point Theory Appl. , 3(2):449–463, 2008.[17] I. Kan. Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin.
Bull. Amer. Math. Soc. (N.S.) , 31(1):68–74, 1994.[18] V. Kleptsyn and P. Saltykov. On C -stable effects of intermingled basins of attractors in classes ofboundary-preserving maps. Trans. Moscow Math. Soc. , vol 17, pages 193–217, 2011.[19] R. Ma˜n´e.
Ergodic theory and differentiable dynamics , volume 8 of
Ergebnisse der Mathematik undihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] . Springer-Verlag, Berlin, Berlin,Heidelberg, 1987.
N THE NON-ROBUSTNESS OF INTERMINGLED BASINS 19 [20] I. Melbourne and A. Windsor. A C ∞ diffeomorphism with infinitely many intermingled basins. ErgodicTheory Dynam. Systems , 25(6):1951–1959, 2005.[21] V. Nit¸ic˘a and A. T¨or¨ok. An open dense set of stably ergodic diffeomorphisms in a neighborhood of anon-ergodic one.
Topology , 40(2):259–278, 2001.[22] A. Okunev. Milnor attractors of circle skew products.
ArXiv e-prints :1508.02132., August 2015.[23] V. Oseledec. A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical sys-tems.
Trudy Moskovskogo Matemati v ceskogo Ob v s v cestva , 19:179–210, 1968.[24] Y. Pesin. Families of invariant manifolds that correspond to nonzero characteristic exponents. Izv.Akad. Nauk SSSR Ser. Mat. , 40(6):1332–1379, 1440, 1976.[25] Y. Pesin. Characteristic Ljapunov exponents, and smooth ergodic theory.
Uspehi Mat. Nauk , 32(4(196)):55–112, 287, 1977.[26] Y. Pesin and Y. Sinai. Gibbs measures for partially hyperbolic attractors.
Ergodic Theory Dynam.Systems , 2(3-4):417–438, 1982.[27] R. Potrie. Partial hyperbolicity and foliations in T . J. Mod. Dyn. T isotopic to Anosov. J. Dynam.Differential Equations , 26(3):805–815, 2014.[29] C. Pugh and M. Shub. Ergodic attractors.
Trans. Amer. Math. Soc. , 312(1):1–54, 1989.[30] F. Rodriguez Hertz, J. Rodriguez Hertz, A. Tahzibi, and R. Ures. Uniqueness of SRB measures fortransitive diffeomorphisms on surfaces.
Comm. Math. Phys. , 306(1):35–49, 2011.[31] F. Rodriguez Hertz, J. Rodriguez Hertz, and R. Ures. A non-dynamically coherent example on T . Ann. Inst. H. Poincar´e Anal. Non Lin´eaire . To appear.[32] R. Ures. Intrinsic ergodicity of partially hyperbolic diffeomorphisms with a hyperbolic linear part.
Proc. Amer. Math. Soc. , 140(6):1973–1985, 2012.[33] Peter Walters.
An introduction to ergodic theory , volume 79 of
Graduate Texts in Mathematics .Springer-Verlag, New York, 1982.[34] L. Young. What are SRB measures, and which dynamical systems have them?
Journal of StatisticalPhysics , 108(5-6):733–754, 2002.
Ra´ul Ures,IMERL, Facultad de Ingenier´ıa, Universidad de la Rep´ublica, CC 30, Montevideo-Uruguay.
E-mail address : [email protected] Carlos H. V´asquez, Instituto de Matem´atica, Pontificia Universidad Cat´olica de Val-para´ıso, Blanco Viel 596, Cerro Bar´on, Valpara´ıso-Chile.
E-mail address ::