Ann. I. H. Poincaré – AN 30 (2013) 463–475
www.elsevier.com/locate/anihpc
On the inverse limit stability of endomorphisms
Pierre Berger
a,∗, Alvaro Rovella
baCNRS-LAGA Institut Galilée, Université Paris 13, France bFacultad de Ciencias, Uruguay
Received 10 November 2011; received in revised form 29 September 2012; accepted 5 October 2012 Available online 23 October 2012
Abstract
We present several results suggesting that the concept of C1-inverse (limit structural) stability is free of singularity theory.
An example of a robustly transitive,C1-inverse stable endomorphism with a persistent critical set is given. We show that every C1-inverse stable, axiom A endomorphism satisfies a certain strong transversality condition(T ). We prove that every attractor–
repeller endomorphism satisfying axiom A and condition(T )isC1-inverse stable. The latter is applied to Hénon maps, rational functions and others. This leads us to conjecture thatC1-inverse stable endomorphisms are exactly those which satisfy axiom A and condition(T ).
©2012 Elsevier Masson SAS. All rights reserved.
Résumé
Nous présentons différents résultats suggérant que le concept deC1-stabilité (structurelle de la limite inverse) est indépendant de la théorie des singularités. Nous décrivons un exemple d’un endomorphisme robustement transitif etC1-stable ayant un ensemble critique persistant. Nous montrons que tout endomorphisme axiome A etC1-stable vérifie nécessairement une certaine condition de transversalité forte(T ). Nous démontrons que tout endomorphisme attracteur–répulseur vérifiant la condition(T )estC1-stable.
Ce dernier résultat est appliqué, entre autres, aux applications de type Hénon et aux fractions rationnelles. Cela nous amène à conjecturer que les endomorphismesC1-stables sont exactement ceux qui vérifient l’axiome A et la condition(T ).
©2012 Elsevier Masson SAS. All rights reserved.
1. Introduction
There exist various concepts of stability for dynamical systems. When dealing with endomorphisms it makes sense to consider the inverse limit which is defined in the sequel. AC1-endomorphismf is aC1-map of a manifoldM into itself, which is not necessarily bijective and which can have a nonemptysingular set(formed by the pointsxs.t.
the derivativeTxf is not surjective). Theinverse limit setoff is the space of the full orbits(xi)i ∈MZoff. The dynamics induced byf on its inverse limit set is the shift. The endomorphismf isC1-inverse limit stableif for every C1perturbationfoff, the inverse limit set off is homeomorphic to the one off via a homeomorphism which conjugates both induced dynamics and isC0-close to the canonical inclusion.
* Corresponding author.
E-mail address:berger.perso@gmail.com(P. Berger).
0294-1449/$ – see front matter ©2012 Elsevier Masson SAS. All rights reserved.
http://dx.doi.org/10.1016/j.anihpc.2012.10.001
When the dynamicsf is a diffeomorphism, the inverse limit set←−Mf is homeomorphic to the manifoldM. The C1-inverse limit stability off is then equivalent to theC1-structural stabilityof f: everyC1-perturbation off is conjugated tof via a homeomorphism ofM.
A great work was done by many authors to provide a satisfactory description ofC1-structurally stable diffeomor- phisms, which starts with Anosov, Smale, Palis, de Melo, Robbin, and finishes with Robinson[14]and Mañé[9].
Such diffeomorphisms are those which satisfy axiom A and the strong transversality condition.
Almost the same description was accomplished for C1-structurally stable flows by Robinson and Hayashi. The inverse limit set of a flow is a one dimensional foliation. The structural stability of a flow is also equivalent to the C1-inverse stability. A flowφisstructurally stableif the foliation induced byφis equivalent to the foliation induced by its perturbation, via a homeomorphism ofMwhich isC0-close to the identity.
The descriptions of the structurally stable maps for smoother topologies (Cr,C∞, holomorphic . . . ) remain some of the hardest, fundamental, open questions in dynamics.
One of the difficulties occurring in the description ofCr-structurally stable smooth endomorphisms concerns the singularities. Indeed, a structurally stable map must display a stable singular set. But there is no satisfactory description of them in singularity theory.
This work suggests that the concept of inverse limit stability does not deal with singularity theory.
For other perspectives, let us mention that the concept of inverse limit stability is an area of great interest for semi-flows given by PDEs, although still at its infancy.
The work of the first author was done during stays at IHES (France), IMPA (Brasil) and Facultad de Ciencias (Uruguay). He is very grateful to these institutes for their hospitality.
2. Statement of the main results
Letf be aC1-map of a compact manifoldMinto itself. Theinverse limitoff is the set
←−Mf :=
x=(xi)i∈Z∈MZ: f (xi)=xi+1∀i∈Z . The set←−Mf is a compact metric space endowed with the metric:
d1(x, y):=
n∈Z
d(xi, yi) 2i ,
withd the Riemannian metric onM. The mapf induces the shift map←f (x)− i =xi+1. We remark that←−Mf is equal toMand←f−is equal tof iff is bijective. Theglobal attractoroff is defined asMf =
n0fn(M). Forj∈Z, let:
πj:x=(xi)i∈MZ→xj∈M.
We note thatπj◦←f−=f◦πj|←−Mf andπj(←−Mf)=Mf.
Two endomorphisms f andf are C1-inverse limit conjugated, if there exists a homeomorphism h from←−Mf onto←−Mf, such that the following equality holds:
h◦←f−=←f−◦h.
Definition 2.1.An endomorphismf isC1-inverse limit stableor simplyC1-inverse stableif everyC1-perturbation foff is inverse limit conjugated tof.
LetKf be a compact,f-invariant subset ofM(f (Kf)⊂Kf). ThenKf ishyperbolicif there exist a sectionEs of the Grassmannian ofT M|Kf and a Riemannian metric satisfying for everyx∈Kf:
• Txf (Es(x))⊂Es(f (x)),
• the action[Txf]induced byf on the quotientsTxM/Es(x)→Tf (x)M/Es(f (x))is invertible,
• Txf|Es(x)<1 and[Txf]−1<1.
We notice that actuallyEs(x)depends only onx0=π0(x). It can be denoted byEs(x0).
On the other hand, there exists a unique continuous family(Eu(x))x of subspacesEu(x)⊂Tx0M, indexed by x∈←K−f :=KfZ∩←−Mf, satisfying:
Tx0f Eu(x)
=Eu←−
f (x)
and Eu(x)⊕Es(x0)=Tx0M.
For >0, the-local stable set ofx∈Kf is:
Ws(x;f )=
y∈M: ∀i0, d
fi(x), fi(y)
, andd
fi(x), fi(y)
→0, i→ +∞
. The-local unstable set ofx∈←K−f is:
Wu(x;←f )− =
y∈←−Mf: ∀i0, d(xi, yi), andd(xi, yi)→0, i→ −∞
.
We will writeWu(x)andWs(x)for Wu(x;←f )− andWs(x;f ) whenever the dynamics involved is clear. Let us justify why we have chosenWs(x)included inMwhereasWu(x)is included in←−Mf. One can prove that (forsmall enough) the local stable set is a submanifold whose tangent space atx equalsEs(x0); however the following is in general not a manifold (not even a lamination).
Ws(x)=Ws(x;←f )− :=π0−1
Ws(x;f )
∩←−Mf.
The-local unstable set is a manifold embedded intoMbyπ0; its tangent space atx0is equal toEu(x). In general the unstable manifold depends on the preorbit: the unstable sets of different orbits inπ0−1(x0)are not necessarily equal.
An endomorphism satisfies (weak)axiom Aif the nonwandering setΩf off is hyperbolic and equal to the closure of the set of periodic points.
In this work, we do not deal withstrong axiom Aendomorphisms which satisfy moreover that the action on each of the basic pieces ofΩf is either expanding or injective. This stronger definition is relevant for structural stability [11,12], but it is conjectured below to be irrelevant for inverse stability. Note also that the present definition of axiom A endomorphisms is more general than the original one of[11]which assumes that the critical set does not intersect the nonwandering set.
We put←Ω−f :=ΩfZ∩←−Mf. In Section4, we will see that←Ω−f is the nonwandering set of←f−. Actually if thef-periodic points are dense inΩf then the←f−-periodic points are dense in←Ω−f. For the sets of the formπN−1(B(x, ))∩←Ω−f, with x∈Ωf, >0 andN∈Z, are elementary open sets of←Ω−f and contain periodic points.
Also ifΩf is hyperbolic the restriction of←f−to←Ω−f is expansive. For theunstable manifold Wu(x)intersects Ws(x)at the unique pointxsinceπ0restricted toWu(x)is a homeomorphism andπ0Wu(x)intersectsWs(π0(x))at the unique pointπ0(x), for everyx∈←Ω−f.
Remark 2.2.Every axiom A endomorphismf has its inverse limit←f−which satisfies axiomA∗in the meaning of[4];
this implies ergodic properties such as the existence of the maximal entropy measure for←f−and so forf. Definition 2.3.The dynamicsf satisfies thestrong transversality condition(T )if:
For alln0,x∈←Ω−f andy∈Ωf, the mapfnrestricted toπ0Wu(x)is transverse toWs(y). In other words, for everyz∈π0Wu(x)∩f−n(Ws(y)):
(T ) Tzfn
Tzπ0Wu(x)
+Tfn(z)Ws(y)=Tfn(z)M.
Definition 2.4.AS endomorphismsare those which satisfy axiom A and the strong transversality condition(T ).
A first result is:
Theorem 2.5.LetMbe a compact manifold andf∈C1(M, M). Iff isC1-inverse stable and satisfies axiom A, then the strong transversality condition holds forf.
The second one concerns the converse:
Definition 2.6.An axiom A endomorphismf isattractor–repellerifΩf is included in the union of two compact hyperbolic setsRf andAf such that there exist:
(a) a neighborhoodVAofAf inMsatisfying
n0fn(VA)=Af, (r) a neighborhoodVRofRf inMf satisfyingRf =
n0f−n(VR)and moreoverf−1(Rf)=Rf. The setRf is called arepellerandAf anattractor.
One can show that iff is an endomorphism whose nonwandering set satisfies(a)and(r)withAf andRf hyper- bolic, thenf is axiom A and so attractor–repeller.
It is not true in general thatΩf=AfRf whenf is attractor–repeller. Indeed, the endomorphismf (x)=x2−1 defined in the real axis has its nonwandering set equal to the union of a (super)-attracting periodic point of period 2 and two repelling fixed points. Moreover, if Rf is defined as the union of the preimages of the fixed points, it is immediate thatf is attractor–repeller.
Theorem 2.7.LetMbe a compact manifold andf ∈C1(M, M). Iff is an attractor–repeller endomorphism which satisfies the strong transversality condition, thenf isC1-inverse stable.
It follows immediately from the theorem of Aoki, Moriyasu and Sumi in [1] that: if an endomorphism f is C1-inverse stable and has no singularities in the nonwandering set, thenf satisfies axiom A.
In Section4, we will study the dynamics and the geometry of the inverse limit of every AS endomorphism. This enables the first author with A. Kocsard to generalize Theorem2.7 to the general case of AS endomorphisms by adding analytical techniques to arguments of Section5. It seems also possible to generalize Aoki–Moruriyasu–Sumi theorem to endomorphisms with singularity. This would complete the following conjecture (sketched in[13]):
Conjecture 2.8. The C1-inverse stable endomorphisms are exactly those which satisfy axiom A and the strong transversality condition(T ).
2.1. Application of Theorem 2.7
Example 2.9(Rational functions).Letf be a rational function of the Riemann sphere. Let us suppose that all its criti- cal points belong to basins of attracting periodic orbits, or equivalently that its Julia set is expanding. By Theorem2.7, f isC1-inverse stable. Note thatC1-perturbations off may have very wild critical set. See[8]for a nice geometrical description of the inverse limit off.
Example 2.10(One-dimensional dynamics and Hénon maps).Kozlovski, Shen and van Strien showed that a(C∞)- generic mapf of the circleS1is attractor–repeller[7], and soC1-inverse limit stable, by Theorem2.7.
Letf(θ, y)=(f (θ )+y,0)be defined on the 2-torusT2which enjoys of a canonical Abelian group structure.
Aside finitely many attracting periodic points, the nonwandering set of f consists of an expanding compact set of f times{0}. This productR is a hyperbolic set forf and a repeller (for the restriction off toMf), as stated in Definition2.6. It follows thatfsatisfies the requirements of Theorem2.7. This implies that ifg∈C1(T2,R)is close to 0, then the inverse limits sets off and of the map:
(θ, y)→
f (θ )+y, g(θ, y) , are conjugated.
For instance, takef (x)=x2+cwithc∈(−2,1/4)attractor–repeller on the one-point compactification ofR. The infinity is an attracting fixed point with basin bounded by the positive fixed pointpoff and its preimage. Letρbe a smooth function with compact support inRand equal to 1 on a neighborhood of[−p, p].
For such a cand thenb small enough, the global attractor of the Hénon map(x, y)→(x2+c+y, bx)of R2, equal to the one of(x, y)→(x2+c+y, ρ(x)·b·x)without the basin of(∞,0), is conjugated to the inverse limit off|[−p, p].
The same example works withf a hyperbolic rational function of the sphere. This generalizes many results in this direction to the wideC1-topology (see[5]which contains other references).
Example 2.11 (C1 robustly transitive, Anosov endomorphism with persistent critical set). Przytycki showed that an Anosov endomorphism without singularities is inverse stable [12]. Latter Quandt generalized this for Anosov endomorphisms, possibly with singularities[13]. These results are consequences of Theorem2.7.
The simplest known example of Anosov endomorphisms is action of linear maps on the quotient R2/Z2, for instance:
A= n 1
1 1 , n∈ {2,3, . . .}.
A constant map is a trivial example of an Anosov endomorphism. Let us construct an example of Anosov map whose singular set is persistently nonempty and whose nonwandering set is the whole manifold.
Begin with a linear mapAof the plane as above. Close to the fixed point one can use linear coordinates to write the map as
λ 0
0 μ ,
where 0< μ <1 andλ >1. Let be a positive constant and letΨ be a nonnegative smooth function such that Ψ (0)=1 andΨ (x)=0 for every|x|> . Assume also thatΨ is an even function having a unique critical point in (−, ). Letϕbe theC1function defined by:ϕ(y)=0 for everyy /∈ [0, ]andϕ(y)=sin(2πy )fory∈ [0, ]. Letf be theC1-endomorphism of the torus equal to
f (x, y)=
λx, μy−Ψ (x)ϕ(y) on the 2-neighborhood of 0 and toAoff.
Letgbe the real functiony→μy−φ(y). There are regular points with different numbers ofg-preimages. The same occurs forf. Consequentlyf has a singular set which is persistently nonempty.
We remark thatf is Anosov. For theA-stable direction is still preserved and contracted; the action ofTf on the stable foliation normal bundle is stillλ-expanding.
Moreover the stable leaves are irrational lines of the torus, uniformly contracted. Given a segmentI contained in a stable leave and a numberA >0, there exists a positive integerk0such that any component off−k(I )has length greater than Afor every kk0. From this it comes that given nonempty open sets U andV,f−k(U ) contains a sufficiently long segment, and thus it intersectsV for every k large. In other words, f is mixing. It follows from Theorem 2.7 that everyC1perturbationfoff is inverse conjugated tof. As a map is transitive if and only if its inverse limit is transitive and projects on the whole manifold, we conclude thatfis transitive.
Example 2.12(Products).Theorem2.7shows also the inverse stability of product of an Anosov endomorphism with an attractor–repeller endomorphism.
Example 2.13.(See Mañé and Pugh[10].) Mañé and Pugh gave an example ofC1-Ω-stable endomorphism for which the singular set persistently intersects an attracting basic piece. Their example is clearly notCr-structurally stable but according to Theorem2.7it isC1-inverse-limit stable.
3. Proof of Theorem2.5
We begin this section with two well known facts of transversality theory.
Claim 3.1.LetN1andN2be two embedded submanifolds ofM.
(i) The set of mapsf ∈C1(M, M)such thatfn|N1is transverse toN2for everyn1is residual.
(ii) If f is a C1 map and f|N1 is not transverse to N2, then there exists a C1 perturbationf off such that f −1(N2)∩N1contains a submanifold whose codimension is less than the sum of the codimensions ofN1andN2. Letf be aC1-inverse stable endomorphism satisfying axiom A. For each small perturbationfoff, leth(f)be the conjugacy between←−Mf and←−Mf.
We will assume by contradiction that the transversality condition fails to be true. This means that there existn0, x∈←Ω−f,y∈Ωf andz∈f−n(Ws(y;f ))∩π0Wu(x;←f )− such that Eq.(T )does not hold. Note first thatz /∈Ωf, by hyperbolicity of the nonwandering set.
Moreover, by density of periodic orbits in←Ω−f after replacingf by a perturbation, we can assume thatxandy are periodic points. To simplify the calculations, we can suppose thatx andy are fixed points by considering an iterate off.
The conjugacyh(f)was asked to be close to the inclusion of←−Mf intoMZ. By expansiveness of←Ω−f, if a per- turbationf is equal tof at the nonwandering setΩf, thenh(f)is equal to the inclusion of←Ω−f. We will produce perturbationsfandfoff that are equal tof onΩf.
The second item of Claim3.1can be used to produce a perturbationfoff such that f −N
Ws y;f
∩π0Wu x;←f−
contains a submanifold of dimension p > u+s−m, whereuis the dimension ofπ0Wu(x),sis the dimension of Ws(x)andmthe dimension of the manifoldM.
On the other hand, the first item of Claim3.1implies that for generic perturbationsfoff, the restriction of(f)k toπ0Wu(x;←f−)is transverse toWs(y;f)for every positive integerk.
If >0 is sufficiently small, the maps f andf are injective restricted to the closures of π0Wu(x;f)and π0Wu(x;f)respectively. This implies that the restrictions ofπ0to the closures ofWu(x;f)andWu(x;f)are homeomorphisms onto their images.
For y∈π0−1({y}), note that π0−1(Ws(y;f)) and π0−1(Ws(y;f)) are equal to the stable sets Ws(y;←f−)and Ws(y;←f−)respectively.
Consequently A:=Ws(y;←f−)∩Wu(x;←f−)contains a manifold of dimensionp whereas A:=Wu(x;←f−)∩ Ws(y;←f−)is a (possibly disconnected) manifold of dimensionu+s−m < p.
We assumed thatf is inverse stable, so the mapφ:=h(f)−1◦h(f)is a conjugacy between←f−and←f−which fixesy andx. Thusφmust embedAintoWu(x;←f−)∩Ws(y,←f−)=
n0
←−
fn(A).
As ←f− is homeomorphism, a manifold of dimension p, contained inA, is embedded by φ into the manifold
n0
←−
fn(A)of dimension less thanp. This is a contradiction.
4. General properties on axiom A endomorphisms
This section is devoted to the study of the geometry and the dynamics of the inverse limit of AS endomorphisms.
Let us first remark that ←Ω−f :=ΩfZ∩←−Mf is also the nonwandering set of←f−. Indeed, an elementary open setV of ←−Mf has the form (
i<NM×U×
i>NM)∩←−Mf =
n∈Zfn−N(U )∩←−Mf, whereU is an open set inM.
Therefore←f−n(V )intersectsV forn >0, ifffn(U )intersectsU.
A compact hyperbolic setK⊂Mhas alocal product structure, if there exists >0 small, such that for every pair of nearby pointsx, y∈←K−f =KZ∩←−Mf, the setWu(x)intersectsWs(x)at a unique point[x, y]whichbelongs to←K−f. Hereis sufficiently small so that in particularWu(x)is embedded byπ0. The following is classical in the diffeomorphism case:
Lemma 4.1.IfKis a basic piece of an axiom A mapf, then←K−f has a local product structure.
Proof. Ifx, yare close enough thenπ0Wu(x;←f )− intersectsπ0Ws(y;←f )− at a unique pointz. Thus for >0 small enough, it holds that for every pair of periodic pointsx, yclose tox, y, the local unstable manifoldπ0Wu(x;←f )− intersectsπ0Ws(y;←f )− at a unique pointz. Asπ0Wu(y;←f )− intersectsπ0Ws(x;←f ), the point− z is nonwandering.
Thuszis nonwandering and also its preimage[x, y]byπ0|Wu(x;←f ).− 2 The existence of a local product structure is useful to shadow:
Lemma 4.2.A hyperbolic compact setK equipped with a local product structure satisfies the shadowing property:
there existsη >0such that everyη-pseudo-orbit(xi)i∈Zin anη-neighborhood ofKis-shadowed by an orbit(yi)i∈Z inK.
Proof. The proof of this lemma is treated as for diffeomorphisms[15, Prop. 8.20]. 2 The following is going to be used many times.
Lemma 4.3.Iff satisfies axiom A, then←−Mf is equal to the union of the unstable manifolds of←Ω−f’s points.
Proof. Every point has itsα-limit set in←Ω−f, and so forn0 large, the points←f−−n(x)are close to←Ω−f. By shadowing we show that such←f−−n(x)belong to local unstable manifolds of←Ω−f’s points. 2
As a consequence of the above lemma and condition(T ), for every AS endomorphismf, everyx∈Ωf,n0, and everyz∈f−n(Ws(x))∩Mf, it holds that:
Tzfn(TzM)+Tfn(z)
Ws(x;f )
=Tfn(z)M. (4.1)
In other words, each iterate off is transverse to each local stable manifold.
4.1. Spectral decomposition and filtration
Letf be aC1-endomorphism of a compact manifoldM, and←f−the associated homeomorphism of the inverse limit←−Mf. Let(Λi)Ni=1be a family of disjoint,←f−-invariant compact subsets of←−Mf.
Afiltration adapted to(Λi)iis a family of open sets(Ui)Ni=0which satisfies the following properties:
(i) ∅ =U0⊂U1⊂ · · · ⊂Ui⊂ · · · ⊂UN=←−Mf, (ii) for everyi,cl(←f (U− i))⊂Ui,
(iii) for every 1iN,Λi=
n∈Z←−
fn(Ui\Ui−1).
We notice that if there exists a filtration adapted to(Λi)i then the limit set←L−f of←f−is included in
iΛi. Given an←f−-invariant, compact subsetΛ, let us define:
Ws(Λ):=
x∈←−Mf: d←−
fn(x), Λ +∞
−−→0
, Wu(Λ):=
x∈←−Mf: d←−
fn(x), Λ −∞
−−→0 .
Given two←f−-invariant, compact subsetsΛandΛ, we putΛΛifWu(Λ)\ΛintersectsWs(Λ)\Λ. We remark that if there exists a filtration adapted to(Λi)i, then the following properties hold:
(a) the limit set←L−f of←f−is included in
iΛi, (b) for alli, j,ΛiΛj impliesi > j.
Actually the two above properties are also sufficient conditions to have a filtration:
Proposition 4.4.(See Theorem12.3 of[15].) Let(Λi)Ni=1be a family of disjoint,←f−-invariant compact subsets of←−Mf. If properties(a)and(b)hold, then there exists a filtration adapted to(Λi)Ni=1.
As for attractor–repellerf, conditions (a) and (b) are satisfied for←R−f ←A−f, it holds:
Corollary 4.5.There exists an open neighborhoodUof←R−f in←−Mf such that:
n0
←−
fn(U )=←R−f, cl←− f−1(U )
⊂U,
n0
←−
fn(U )=←−Mf\Af.
In general, for an axiom A endomorphismf, letN
i=1Ωi be the splitting of Ωf into maximal transitive sets.
These sets are closed andf-stable:f (Ωi)=Ωi, for everyi. Put←Ω−i:=ΩiZ∩←−Mf. We notice that←Ω−f =
i
←−
Ωi. The families(Ωi)i and(←Ω−i)i are called thespectral decompositionof respectivelyΩf and←Ω−f.
1 Actually the cited theorem asks←f−to be a homeomorphism of amanifold, but this fact is useful uniquely to choose(Ui)iamong the submani- folds with boundary.
Proposition 4.6.For every AS endomorphismf, we can index(←Ω−i)i such that for alli, j,←Ω−i←Ω−jimpliesi > j. Proof. The proof is done as for diffeomorphisms: it follows from the fact that if←Ω−i←Ω−j then there exist periodic pointsp∈←Ω−i andq∈←Ω−jsuch thatWu(p)intersectsWs(q)and that for anyr∈←Ω−j,Wu(q)intersectsWs(r). 2 Corollary 4.7.There exists a filtration adapted to the spectral decomposition(←Ω−i)iof every AS endomorphism.
4.2. Stratifications of the inverse limit by laminations
We are going to study the geometry of the inverse limit←−Mf of AS endomorphisms. This set is in general not a manifold not even a lamination (see for instance Example2.10). However we are going to stratify it by laminations.
This will be suitable to construct the conjugacy in the attractor–repeller case. Let us recall some elements of the lamination theory applied to hyperbolic dynamical systems.
Alaminationis a secondly countable metric spaceLlocally modeled on open subsetsUi of products ofRnwith locally compact metric spacesTi (via homeomorphisms calledcharts) such that the changes of coordinates are of the form:
φij=φj◦φi−1:Ui⊂Rn×Ti→Uj⊂Rn×Tj, (x, t)→
g(x, t ), ψ (x, t ) ,
where the partial derivative w.r.t.xofgexists and is a continuous function of bothx andt, alsoψis locally constant w.r.t.x. A maximal atlasLof compatible charts is alamination structureonL.
Aplaqueis a component ofφ−i 1(Rn× {t})for a chart φi andt ∈Ti. Theleaf of x∈L is the union of all the plaques which containx. A leaf has a structure of manifold of dimensionn. Thetangent spaceTLofLis the vector bundle overLwhose fiberTxLatx∈Lis the tangent space atxof its leaf.
Proposition 4.8. Let K be a hyperbolic compact, invariant set of an endomorphismf which has a local product structure. Let ←K−:=KZ∩←−Mf. Then the local unstable manifolds (Wu(x))x∈←K− form the plaques of a lamination onWu(←K)− :=
x∈←K−Wu(x). Moreover the local stable manifolds(Ws(x))x∈K form the plaques of a lamination on Ws(K):=
x∈KWs(x).
Proof. The proofs of both statements are similar and so, we shall only show the one regardingWu(←K). Let us express− some charts of neighborhoods of anyx∈←K−that span the laminar structure onWu(←K). By the local product structure,− for everyy∈←K−close tox, the intersection ofWu(y)withWs(x)is a pointt= [y, x]in←K. Also we can find a family− of homeomorphisms(φt)t which depend continuously ontand sendWu(t)ontoRd. We notice that the map:
y→
φt(y), t
∈Rd×Ws(x)
is a homeomorphism which is a chart of lamination. 2
Corollary 4.9.LetΩi be a basic piece of an AS endomorphismf. Then the unstable manifolds of points in←Ω−i form the leaves of a lamination onWu(←Ω−i).
Proof. We recall that there are open setsUiandUi−1such that:
←−
Ωi=
n
←−
fn(Ui\Ui−1), cl←−
f (Ui)
⊂Ui, cl←−
f (Ui−1)
⊂Ui−1. First let us notice thatWu(←Ω−i)⊂Ui. By←f−-invariance ofWu(←Ω−i), it is included in
n∈Z←−
fn(Ui).
Letx∈Wu(←Ω−i), there existsN0 such that←f−−N(x)is included in the interior ofUic−1. It is also the case for a neighborhoodU ofxinWu(←Ω−i):
f−N(U )⊂Uic−1∩Wu(←Ω−i)⊂Uic−1∩
n∈Z
←−
fn(Ui).
As the decreasing sequence of compact sets(←f−−p(Uic−1)∩
n∈Z←−
fn(Ui))p0 converges to ←Ω−i in the Hausdorff topology, forplarge,←f−−p(U )lies in a small neighborhood of←Ω−i. By shadowing,←f−−p(U )is included inWu(←Ω−i).
As←f−pis a homeomorphism, we can push forward the lamination structure onU. 2
The above proof can be also applied for attractor–repeller endomorphisms. Indeed, it uses only the existence of a filtration and of the local product structure (in order to shadow).
Let us recall that by Lemma4.3, iff is attractor–repeller or AS, then the unstable manifolds of the nonwandering set form a partition of the inverse limit←−Mf. Thus from the results above, the inverse limit←−Mf is stratified by a finite number of laminations, the leaves of which are unstable manifolds.
As a consequence of the strong transversality, it follows also a similar partition by stable manifolds of a neighbor- hood ofMf. However, as this construction holds on the manifoldMand not on the inverse limit, let us first recall some general definitions and facts about transversality.
We recall that a continuous mapgfrom a laminationLto a manifoldM isof classC1if its restriction to every plaque ofL is a C1 map of manifolds and the induced map T g:TL→T M is continuous: the restriction Txg: TxL→Tg(x)Mdepends continuously onx even transversally.
For instance the restriction ofπ0to any of the above laminations by unstable manifold is of classC1.
LetLbe a lamination embedded intoM. The laminationListransverse toLviagif for everyx∈Lsuch that g(x)belongs toL, the following equality holds:
T g(TxL)+Tg(x)L=Tg(x)M.
Claim 4.10.There exists a laminationLgL onL∩g−1(L)the plaques of which are intersections ofL-plaques withg-preimages ofL-plaques.
Proof. Let us construct a chart ofLgLfor distinguished open sets which coverL∩g−1(L). Letx∈L∩g−1(L), letφ:U→Rd×T be anL-chart of a neighborhoodUofx and letφ:U→Rd×Tbe anL-chart of a neighbor- hoodUofg(x).
For eacht∈T, letT(t)be the set oft∈Ts.t.g◦φ−1(Rd×{t})intersectsφ −1(Rd×{t}). LetPt,tbe theg-pull back of this intersection. We notice thatPt,t depends continuously on(t, t)∈
t∈TT(t)as aC1-manifold ofM.
By restrictingU,Pt,t is diffeomorphic toRd+d−n, via a mapφt,t :Pt,t →Rd+d−n which depends continuously ont, t. This provides a chart:
U∩g−1 U
→Rd+d−n×
t∈T
T(t), x∈Pt,t→
φt,t(x), t, t
. 2
The above claim is useful for the following proposition:
Proposition 4.11. For every AS endomorphism f, for every basic piece Ωi, there exists a neighborhood Vi of Ws(Ωi)∩Mf which supports a lamination by stable manifolds.
If f is furthermore attractor–repeller, then a neighborhood of Ws(Af)∩Mf supports a lamination by stable manifolds.
Proof. We are going to prove only the second statement on attractor–repeller endomorphismsf since the proof of the first statement is a simple combination of it with the argument using filtration of Corollary4.9.
We recall that by Proposition4.8, the setWs(Af)supports a structure of lamination by local stable manifolds.
On the other hand, the manifoldMis a lamination formed by a single leaf. We wish to use Claim4.10withg=fn, L:=Ws(Af)andL=M.
Howeverfnis not necessarily transverse toWs(Af)ofMf, and soWs(Af)=
n0f−n(Ws(Af))is in general not endowed with a structure of lamination.
Nevertheless, by Eq. (4.1), transversality occurs at a neighborhoodUnofMf ∩f−n(Ws(Af)). This implies the existence of a structure of laminationLs onU∩Ws(Af), withU:=
n0Un. 2
