On the inverse limit stability of endomorphisms

There exists 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. A $C^{1}$-endomorphism $f$ is a $C^{1}$-map of a manifold $M$ into itself, which is not necessarily bijective and which can have a nonempty singular set (formed by the points $x$ s.t. the derivative $T_{x}f$ is not surjective). The inverse limit set of $f$ is the space of the full orbits $(x_{i})_{i}\in M^{\mathbb{Z}}$ of $f$. The dynamics induced by $f$ on its inverse limit set is the shift. The endomorphism $f$ is $C^{1}$-inverse limit stable if for every $C^{1}$ perturbation $f^{\prime}$ of $f$, the inverse limit set of $f^{\prime}$ is homeomorphic to the one of $f$ via a homeomorphism which conjugates both induced dynamics and close to the canonical inclusion.

When the dynamics $f$ is a diffeomorphism, the inverse limit set $M_{f}$ is homeomorphic to the manifold $M$. The $C^{1}$-inverse limit stability of $f$ is then equivalent to the $C^{1}$-structural stability of $f$: every $C^{1}$-perturbation of $f$ is conjugated to $f$ via a homeomorphism of $M$.

A great work was done by many authors to provide a satisfactory description of $C^{1}$-structurally stable diffeomorphisms, which starts with Anosov, Smale, Palis [PS70], de Melo, Robbin, and finishes with Robinson [Rob76] and Mañé [Mañ88]. Such diffeomorphisms are those which satisfy axiom A and the strong transversality condition.

Almost the same description was accomplished for $C^{1}$-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 $C^{1}$-inverse stability. A flow $\phi$ is structurally stable if the foliation induced by $\phi$ is equivalent to the foliation induced by its perturbation, via a homeomorphism of $M$ which is $C^{0}$-close to the identity.

The descriptions of the structurally stable maps for smoother topologies ($C^{r}$, $C^{\infty}$, holomorphic…) remain some of the hardest, fundamental, open questions in dynamics.

One of the difficulties occurring in the description of $C^{r}$-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.

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.

Let $f$ be a $C^{1}$-map of a compact manifold $M$ into itself.

The inverse limit of $f$ is the set $M_{F}:=\{\underline{x}\in M^{\mathbb{Z}}:\;f(x_{i})=x_{i+1}\;\forall i\in\mathbb{Z}\}$, where $M^{\mathbb{Z}}$ is the space of sequences $\underline{x}=(x_{i})_{i\in\mathbb{Z}}$. The subset $M_{F}$ endowed with the induced product topology is compact. The map $f$ induces the shift map $F(\underline{x})_{i}=x_{i+1}$. We remark that $M_{F}$ is equal to $M$ and $F$ is equal to $f$ if $f$ is bijective. The global attractor of $f$ is defined as $M_{f}=\cap_{n\geq 0}f^{n}(M)$. For $j\in\mathbb{Z}$, let:

We note that:

Also a point $z$ belongs to $M_{f}$ if and only if $\pi_{0}^{-1}(\{z\})$ is not empty. Although $\pi_{j}$ depends on $f$, this will be not emphasized by an explicit notation.

Two endomorphisms $f$ and $f^{\prime}$ are $C^{1}$-inverse limit conjugated, if there exists a homeomorphism $h$ from $M_{F}$ onto $M_{F^{\prime}}$, such that the following equality holds:

Let $K_{f}$ be a compact, $f$-invariant subset of $M$ ($f(K_{f})\subset K_{f}$). Then $K_{f}$ is hyperbolic if there exists a section $E^{s}$ of the Grassmannian of $TM|K_{f}$ and $N>0$ satisfying for every $x\in K_{f}$:

• •

  $T_{x}f(E^{s}(x))\subset E^{s}(f(x))$,

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  the action $[Tf]$ induced by $f$ on the quotients $T_{x}M/E^{s}(x)\rightarrow T_{f(x)}M/E^{s}(f(x))$ is invertible,

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  $\|T_{x}f^{N}|E^{s}(x)\|<1$,

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  $\|([Tf]^{N})^{-1}\|<1$.

We notice that actually $E^{s}(\underline{x})$ depends only on $x_{0}=\pi_{0}(\underline{x})$. It can be denoted by $E^{s}(x_{0})$.

On the other hand, there exists a unique continuous family $\big{(}E^{u}(\underline{x})\big{)}_{\underline{x}}$ of subspaces $E^{u}(\underline{x})\subset T_{x_{0}}M$, indexed by $\underline{x}\in K_{F}:=K_{f}^{\mathbb{Z}}\cap M_{F}$, satisfying:

For $\epsilon>0$, the $\epsilon$-local stable set of $x\in K_{f}$ is:

The $\epsilon$-local unstable set of $\underline{x}\in K_{F}$ is:

Let us justify why we have chosen $W^{s}_{\epsilon}(x)$ included in $M$ whereas $W^{u}_{\epsilon}(\underline{x})$ is included in $M_{F}$. One can prove that (for $\epsilon$ small enough) the local stable set is a submanifold whose tangent space at $x$ equals $E^{s}(x_{0})$; however its preimage $W^{s}_{\epsilon}(\underline{x};f)$ by $\pi_{0}$ is in general not a manifold (not even a lamination in general). The local unstable set is a manifold embedded into $M$ by $\pi_{0}$; its tangent space at $x_{0}$ is equal to $E^{u}(\underline{x})$. In general the unstable manifold depends on the preorbit: the unstable sets of different orbits in $\pi^{-1}(x_{0})$ are not necessarily equal.

An endomorphism satisfies (weak) axiom A if the nonwandering set $\Omega_{f}$ of $f$ is hyperbolic and equal to the closure of the set of periodic points.

In this work, we do not deal with strong axiom A endomorphisms which satisfy moreover that the action on each of the basic pieces of $\Omega_{f}$ is either expanding or injective. This stronger definition is relevant for structural stability [Prz77], but it is conjectured below to be irrelevant for inverse stability.

We put $\Omega_{F}:=\Omega_{f}^{\mathbb{Z}}\cap M_{F}$. Actually if the $f$-periodic points are dense in $\Omega_{f}$ then the $F$-periodic points are dense in $\Omega_{F}$. For the sets of the form $\pi_{N}^{-1}(B(x,\epsilon))\cap\Omega_{F}$, with $x\in\Omega_{f}$, $\epsilon>0$ and $N\in\mathbb{Z}$, are elementary open sets of $\Omega_{F}$ and contain periodic points.

Also if $\Omega_{f}$ is hyperbolic the restriction of $F$ to $\Omega_{F}$ is expansive. For the $\epsilon$ unstable manifold $W^{u}_{\epsilon}(\underline{x})$ intersects $W^{s}_{\epsilon}(\underline{x})$ at the unique point $\underline{x}$ since $\pi_{0}$ restricted to $W^{u}_{\epsilon}(\underline{x})$ is a homeomorphism and $\pi_{0}(W^{u}_{\epsilon}(\underline{x}))$ intersects $W^{s}_{\epsilon}(\pi_{0}(\underline{x}))$ at the unique point $\pi_{0}(\underline{x})$, for every $\underline{x}\in\Omega_{F}$.

A first result is:

The second one concerns the converse:

It follows immediately from the Theorem of Aoki-Moriyasu-Sumi in [AMS01] that:

If an endomorphism $f$ is $C^{1}$-inverse stable and has no singularities in the nonwandering set, then $f$ satisfies axiom A.

Hoping to generalize this result and Theorem 2.5, we propose the following conjecture (vaguely written in [Qua88]):

We begin this section with two well known facts of transversality theory.

Let $f$ be a $C^{1}$-inverse stable endomorphism satisfying axiom A. For each small perturbation $f^{\prime}$ of $f$, let $h(f^{\prime})$ be the conjugacy between $M_{F}$ and $M_{F^{\prime}}$.

We will assume by contradiction that the transversality condition fails to be true. This means that there exist $n\geq 0$, $\underline{x}\in\Omega_{F}$, $y\in\Omega_{f}$ and $z\in\pi_{0}\big{(}W^{u}_{\epsilon}(\underline{x};F)\big{)}\cap f^{-n}\big{(}W^{s}_{\epsilon}(y;f)\big{)}$ such that Equation $(T)$ does not hold. Note first that $z\notin\Omega_{f}$, by hyperbolicity of the nonwandering set.

Moreover, by density of periodic orbits in $\Omega_{F}$ after replacing $f$ by a perturbation, we can assume that $\underline{x}$ and $y$ are periodic points. To simplify the calculations, we can suppose that $\underline{x}$ and $y$ are fixed points by considering an iterate of $f$.

The conjugacy $h(f^{\prime})$ was asked to be close to the inclusion of $M_{F}$ into $M^{\mathbb{Z}}$. By expansiveness of $\Omega_{F}$, if a perturbation $f^{\prime}$ is equal to $f$ at the nonwandering set $\Omega_{f}$, then $h(f^{\prime})$ is equal to the inclusion of $\Omega_{F}$. We will produce perturbations $f^{\prime}$ and $f^{\prime\prime}$ of $f$ that are equal to $f$ on $\Omega_{f}$.

The second item of Claim 3.1 can be used to produce a perturbation $f^{\prime}$ of $f$ such that

contains a submanifold of dimension $p>u+s-m$, where $u$ is the dimension of $\pi_{0}(W^{u}_{\epsilon}(\underline{x}))$, $s$ is the dimension of $W^{s}_{\epsilon}(x)$ and $m$ the dimension of the manifold $M$.

On the other hand, the first item of Claim 3.1 implies that for generic perturbations $f^{\prime\prime}$ of $f$, the restriction of $(f^{\prime\prime})^{k}$ to $\pi_{0}(W^{u}_{\epsilon}(\underline{x};F^{\prime\prime}))$ is transverse to $W^{s}_{\epsilon}(y;f^{\prime\prime})$ for every positive integer $k$.

If $\epsilon>0$ is sufficiently small, the maps $f^{\prime}$ and $f^{\prime\prime}$ are injective restricted to the closures of $\pi_{0}\big{(}W^{u}_{\epsilon}(\underline{x};f^{\prime})\big{)}$ and $\pi_{0}\big{(}W^{u}_{\epsilon}(\underline{x};f^{\prime\prime})\big{)}$ respectively. This implies that the restrictions of $\pi_{0}$ to the closures of $W^{u}_{\epsilon}(\underline{x};f^{\prime})$ and $W^{u}_{\epsilon}(\underline{x};f^{\prime\prime})$ are homeomorphisms onto their images.

For $\underline{y}\in\pi_{0}^{-1}(y)$, note that $\pi_{0}^{-1}(W^{s}(y;f^{\prime}))$ and $\pi_{0}^{-1}(W^{s}(y;f^{\prime\prime}))$ are equal to the stable sets $W^{s}(\underline{y};F^{\prime})$ and $W^{s}(\underline{y};F^{\prime\prime})$ respectively.

Consequently $A^{\prime}:=W^{s}(\underline{y};F^{\prime})\cap W^{u}_{\epsilon}(\underline{x};F^{\prime})$ contains a manifold of dimension $p$ whereas $A^{\prime\prime}:=W^{u}_{\epsilon}(\underline{x};F^{\prime\prime})\cap W^{s}(y;F^{\prime\prime}))$ is a (possibly disconnected) manifold of dimension $u+s-m<p$.

We assumed that $f$ is inverse stable, so the map $\phi:=h(f^{\prime\prime})^{-1}\circ h(f^{\prime})$ is a conjugacy between $F^{\prime}$ and $F^{\prime\prime}$ which fixes $\underline{y}$ and $\underline{x}$. Thus $\phi$ must embed $A^{\prime}$ into $W^{u}(\underline{x};F^{\prime\prime})\cap W^{s}(\underline{y},F^{\prime\prime})=\cup_{n\geq 0}F^{\prime\prime n}(A^{\prime\prime})$.

As $F^{\prime\prime}$ is homeomorphism, a manifold of dimension $p$, contained in $A^{\prime}$, is embedded by $\phi$ into the manifold $\cup_{n\geq 0}F^{\prime\prime n}(A^{\prime\prime})$ of dimension less than $p$. This is a contradiction.