Posted by: matheuscmss | May 6, 2010

Axiom A versus Newhouse phenomena for Benedicks-Carleson toy models

A few days ago, Carlos Gustavo (Gugu) Moreira, Enrique Pujals and I uploaded to Arxiv our paper Axiom A versus Newhouse phenomena for Benedicks-Carleson toy models.

In a previous post, I gave a rough outline (and a link to a short announcement already published by Oberwolfach Reports) of this paper. In particular, we discussed Smale’s conjecture and the main known obstruction to the validity of its analogous among {C^2} diffeomorphisms of surfaces, namely, Newhouse phenomena (a local mechanism to create {C^2} robust homoclinic tangencies using {C^{1+\alpha}}-stable intersections of dynamically defined Cantor sets).

However, after taking a look at this previous post, I found out that, although we had three sources of motivation for this project, I have mentioned only two of them. More precisely, our strategy can be resumed in a few words as:

  • Try to extend (to the two-dimensional setting) a “modern” proof nicely sketched in the excelent book of de Melo and van Strien of a theorem of M. Jakobson on the {C^1}-denseness of Axiom A among unimodal maps of the interval (i.e., Smale’s conjecture in one dimension);
  • During this process, apply Gugu’s theorem about the non-existence of {C^1}-stable intersections of dynamical Cantor sets to get rid of the eventual tangencies (our potential enemies [in view of Newhouse phenomena]) obstructing Axiom A;
  • Because it isn’t clear how to perform the previous scheme starting with arbitrary {C^1} diffeomorphisms of surfaces (since a priori robust tangencies aren’t the sole enemies), we restrict our considerations to the so-called Benedicks-Carleson toy models: they are a simplified version of Henon dynamics with sufficiently simple geometry (so that our ideas could be tested without entering a huge number of technical troubles) but sharing part of the dynamical complications of (true) Henon maps (e.g., they exhibit some sort of Newhouse phenomena). See our paper for the precise definition of these toy models.

Concerning these three items, the quoted post touches upon the last two topics. Therefore, besides making some propaganda for the preprint :D, I decided to take the opportunity to talk about Jakobson’s theorem and how it connects to the strategy of our paper.

{C^1}-denseness of Axiom A among unimodal maps

Theorem 1 (Jakobson) Axiom A is {C^1}-dense among the set of unimodal maps of the interval {I=[-1,1]}.

Remark 1 Recall that a map {f} of the interval is called unimodal if {f} has exactly one critical point {c}.

The original argument of Jakobson is a little bit long because he deals with multimodal maps (i.e., intervals maps with several critical points) and his criterion of uniform hyperbolicity is somewhat technical. For these reasons, as we already pointed out, we will adopt here a modern strategy outlined in the excelent book of W. de Melo and S. van Strien: using Mañé’s criterion of hyperbolicity for intervals maps (certainly not available at the time of writing of Jakobson’s paper), we know that uniform hyperbolicity holds for the set of points whose orbits stay “away” from the critical point. Thus, it remains only to analyze the dynamics nearby the criticality. At this point, the idea is the following: after a {C^1}-small perturbation, it is possible to put the orbit of the critical point inside the basin of attraction of a sink (i.e., an attracting periodic point). Once we get this fact, the proof of Jakobson’s theorem is complete since we are showing that the points of the non-wandering set should stay far away from the critical point. Indeed, if the orbit of a point of the non-wandering set is very close to the critical point, it falls into the basin of a sink (since we are assuming that the critical point is absorbed by a sink). In particular the orbit of this point is wandering, a contradiction. By Mañé’s criterion, it follows that the non-wandering set is the union of finitely many sinks and a compact invariant hyperbolic set (i.e., the map is Axiom A).

Remark 2 Usually, Axiom A requires that the periodic points are dense in the non-wandering set and the non-wandering set is hyperbolic. However, since the endomorphisms considered above are not invertible, generally speaking the non-wandering set is not invariant by backward iteration, so that a slight modification of the Axiom A is needed.

Bearing this plan in mind, let us begin the proof of Jakobson’s {C^1}-density result. We begin with a fundamental criterion of hyperbolicity due to Mañé:

Theorem 2 (Mañé) Let {f} be a {C^2} endomorphism of the interval {I} and {U} be an open neighborhood of the set {C(f)} of critical points of {f}. Denote by {\Sigma(f)} the union of the basins of attractions of the sinks of {f}. It holds:

  • any periodic orbit of {f} inside {I-U} with sufficiently high period is a source;
  • if every periodic point of {f} inside {I-U} is a source, then there are some constants {0<\lambda<1} and {C>0} such that {\|Df^n(x)\|\geq C\lambda^n} for any {\{x,f(x),\dots,f^n(x)\}\subset  I-(U\cup\Sigma(f))}.

Corollary 3 (Mañé) Let {f} be a {C^2} endomorphism of the interval {I} and {\Lambda\subset I} be a compact {f}-invariant set. Assume that all periodic point of {\Lambda} are sources and {\Lambda} does not contain critical points of {f}. Then, {\Lambda} is hyperbolic, i.e., there are constants {0<\lambda<1} and {C>0} such that {\|Df^n(x)\|\geq  C\lambda^n} for all {n\geq 0} and {x\in\Lambda}.

Remark 3 The reader should pay attention to the {C^2} regularity hypothesis in the statements above (which is a necessary assumption).

Keeping these results in our toolbox, we are ready to begin the proof of the {C^1} density of Axiom A among unimodal maps. Let {f} be a {C^1} unimodal map of the interval {I=[-1,1]}. Without loss of generality, we can suppose that:

  • {f} is {C^2}
  • {0} is the critical point of {f} and
  • {f} is Kupka-Smale (all periodic points are hyperbolic).

For sake of simplicity, we denote by {\mathcal{D}^2} the set of endomorphisms of {I=[-1,1]} verifying the three conditions above.

Proposition 4 For a typical (Baire generic) {f\in\mathcal{D}^2}, the critical point {0} is recurrent or it falls into the basin of some sink.

Proof: For each {\varepsilon>0}, let {R(\varepsilon)} be the set of {g\in\mathcal{D}^2} such that either {g^n(0)\in  (-\varepsilon,\varepsilon)} for some {n\geq  1} or {0} falls into the basin of a sink of {g}. We claim that {R(\varepsilon)} is open and dense (for every {\varepsilon>0}). Indeed, since {R(\varepsilon)} is clearly open, our task is reduced to show that {R(\varepsilon)} is dense.

Given {\varepsilon>0} and {g\in\mathcal{D}^2-R(\varepsilon)}, consider

\displaystyle  \Lambda_\varepsilon(g):=\bigcap\limits_{n\geq 0}  g^{-n}(I-(-\varepsilon,\varepsilon)) - \Sigma(g).

Note that {\Lambda_\varepsilon(g)} is a compact invariant set without critical points and sinks. Since {g} is Kupka-Smale, every periodic point inside {\Lambda_\varepsilon(g)} must be a source, so that the corollary 3 implies that {\Lambda_\varepsilon(g)} is hyperbolic, i.e., for some {0<\lambda<1}, {C>0}, we have {\|Df^n(x)\|\geq  C\lambda^n} for all {x\in\Lambda_\varepsilon(g)}. In particular, {\Lambda_\varepsilon(g)} is a Cantor set (i.e., a compact set with empty interior). In fact, if the interior of {\Lambda_\varepsilon(g)} were non-empty, say {J\subset\Lambda_\varepsilon(g)} where {J} is a non-trivial, it would follow that {g^n(J)\subset\Lambda_\varepsilon(g)} so that the condition of hyperbolicity implies {\ell(g^n(J))\geq  C\lambda^n\ell(J)} for all {n\geq  1}. Thus, {\ell(g^n(J))\rightarrow\infty} when {n\rightarrow\infty}, a contradiction with {g^n(J)\subset I} and {\ell(I)=2}. On the other hand, since {g\notin R(\varepsilon)}, it follows that {g(0)\in\Lambda_\varepsilon(g)}.

At this point, we use the following very simple idea: because {\Lambda_\varepsilon(g)} is a Cantor set, the condition {g(0)\in\Lambda_\varepsilon(g)} can not be generic (so that it can be destroyed by small perturbation). More precisely, we consider {f\in\mathcal{D}^2} a {C^1}-small perturbation of {g} supported on {(-\varepsilon,\varepsilon)} such that {f(0)\notin\Lambda_\varepsilon(g)}. We claim that {f\in R(\varepsilon)}. Indeed, since {f} coincides with {g} outside {(-\varepsilon,\varepsilon)}, we get

\displaystyle \bigcap\limits_{n\geq  0}f^{-n}(I-(-\varepsilon,\varepsilon)) = \bigcap\limits_{n\geq 0}  g^{-n}(I-(-\varepsilon,\varepsilon)).

Thus, if {f\notin R(\varepsilon)}, we have

\displaystyle f(0)\in \bigcap\limits_{n\geq  0}f^{-n}(I-(-\varepsilon,\varepsilon)) = \bigcap\limits_{n\geq  0}g^{-n}(I-(-\varepsilon,\varepsilon)).

Hence, using that {f(0)\notin\Lambda_\varepsilon(g)} by construction, we obtain {f(0)\in\Sigma(g)}, that is, {f(0)} is attracted by a sink of {g}. However, since the orbit of {f(0)} under {f} (and consequently {g}) never touches the interval {(-\varepsilon,\varepsilon)}, one sees that the orbit of sink of {g} attracting {f(0)} belongs to {I-(-\varepsilon,\varepsilon)}. Therefore, the sink of {g} attracting {f(0)} is also a sink of {f}. In other words, {0\in  \Sigma(f)}, i.e., {f\in  R(\varepsilon)}, an absurd. This shows that {R(\varepsilon)} is open and dense for any {\varepsilon>0}.

Finally, we complete the proof of the proposition by taking the residual set {R:=\bigcap\limits_{n\in\mathbb{N}} R(1/n)} of {\mathcal{D}^2}. Clearly, any {f\in  R} verifies the statement of the proposition. This finishes the proof. \Box

Remark 4 The argument used in the proof of this proposition does not rely on the {C^1}-topology so that a similar statement holds with respect to the {C^r}-topology for any {r\geq 2}.

Up to now, we do not used the {C^1}-topology in our discussion. However, the next proposition strongly relies on the {C^1}-topology and it is the main obstruction for an extension of Jakobson’s result to the {C^2}-topology, for instance.

Proposition 5 (Flatness perturbation) Let {f\in\mathcal{D}^2} such that {0} is recurrent, i.e., {0\in\omega(0)}. Then, there exists an arbitrarily small {C^1}-perturbation {g\in\mathcal{D}^2} of {f} such that {0} belongs to the basin of attraction of a sink of {g}, i.e., {0\in\Sigma(g)}.

Proof: Since the {\varepsilon-\delta} management of this argument is a little bit hard (while the idea behind it is quite clear), we’ll just sketch the proof of the proposition. We start by selecting a large integer {n} such that {f^n(0)} is very close to {0} and {|f^k(0)|=d(f^k(0),0)>d(f^n(0),0)=|f^n(0)|} for all {0<k<n}.

Denote by {J=[f^n(0),0]}. Since {0\in\omega(0)}, we can made {J} arbitrarily small and we can select {m>n} such that {f^m(0)\in J}. For sake of convenience, we take {m>n} minimal for the property {f^m(0)\in J}. We have two possibilities:

  • (a) {f^m(0)} is more close to {0} than {f^n(0)}, i.e., {d(f^m(0), 0) \leq d(f^m(0),f^n(0))};
  • (b) {f^m(0)} is more close to {f^n(0)} than {0}, i.e., {d(f^m(0),f^n(0))\leq d(f^m(0), 0)}.

In the first case (a), we apply a “Closing Lemma” argument: take {z} to be the middle point between {f^m(0)} and {f^n(0)} and {U} a small open neighborhood of {[z,0]} with {f^k(0)\notin\overline{U}} for all {0<k<m}. See the figure below.

In this situation, we modify {f} inside {U} as follows: denoting by {c_m=f^m(0)}, take {h} {C^1}-close to {f} so that {h(c_m)=f(0)}, {h|U} has an unique critical point at {c_m} and {h=f} outside {U}. It follows that {h} is unimodal with critical point {c_m} such that {c_m} is periodic (the fact that the recurrent point {c_m} becomes periodic justifies the name “Closing Lemma” for this argument). By a {C^1}-perturbation of {h}, we obtain {g} verifying the conclusion of the proposition (in fact, it turns out that here the critical point itself is a super-attracting sink).

Finally, in the second case (b), we introduce the “flatness perturbation”: take {z} to be the middle point between {0} and {f^m(0)} and {U} a small open neighborhood of {[f^n(z),f^m(z)]} such that {f^k(0)\notin\overline{U}} for all {0<k<m} with {k\neq n}. See the figure below.

In this context, we modify {f} as follows. Take {h} {C^1}-close to {f} so that {h} is constant on {[f^n(0),f^m(0)]} and {h(f^n(0))=f(f^n(c))}. Observe that this perturbation can be done (only) in the {C^1}-topology since we know that {0} is a critical point of {f} and {f^n(0), f^m(0)} are fairly close to {0}. Next, note that {f^n(0)} is a periodic point of {h} with period {m-n}. Moreover, {f^n(0)} is a super-attracting sink of {h} (because {h} is constant on {[f^n(0),f^m(0)]} and thus {h'(f^n(0))=0}). This allows us to make a {C^1}-perturbation {g} of {h} so that {g} is unimodal with derivative {g'} almost zero (but never vanishing) on {U} and {g} possesses a sink whose basin contains {f^n(0)=g^n(0)}. See the figure below.

Flatness perturbation -- Case (b).

Consequently, {g\in\mathcal{D}^2} is {C^1}-close to {f} and {0\in\Sigma(g)}. This completes the proof. \Box

Once these two propositions are proved, Jakobson’s theorem follows directly. Indeed, given {f} a {C^1} endormorphism of the interval {I}, we can assume that {f\in\mathcal{D}^2} (as discussed before). Next, we approximate {f} by a “typical” {g\in\mathcal{D}^2} so that the proposition 4 says that either {0\in \Sigma(g)} or {0\in\omega(0)}.

If {0\in\Sigma(g)}, say the critical point {0} falls by iteration into the basin of a sink {p} of {g}), we see that there exists {\varepsilon>0} such that the neighborhood {(-\varepsilon,\varepsilon)} falls into the basin of attraction of the sink {p}. On the other hand, since {g\in\mathcal{D}^2}, we know that {\Lambda_\varepsilon(g)=\bigcap\limits_{n\geq  0}g^n(I-(-\varepsilon,\varepsilon))} is the union of an uniformly expanding hyperbolic set and a finite number of sinks. In particular, we get that {g} is an Axiom A endomorphism arbitrarily {C^1}-close to {f}.

If {0\in\omega(0)} (i.e., the critical point {0} is recurrent), we apply the proposition 5 to get an endomorphism {h\in\mathcal{D}^2} arbitrarily {C^1}-close to {g} so that {0\in\Sigma(h)}. Therefore, the discussion of the previous paragraph gives us some Axiom A endomorphism {k} arbitrarily {C^1}-close to {h}.

Thus, in any case, {f} can be {C^1}-approximated by an Axiom A endomorphism. This completes the proof of Jakobson’s theorem.

Few comments on {C^1}-denseness of Axiom A for Benedicks-Carleson toy models

As we saw in the previous section, the “modern” proof of Jakobson’s theorem has two ingredients: Mañé’s hyperbolicity criterion and adequate perturbations to force critical points to fall into the basins of sinks. Of course, in the two-dimensional setting of Benedicks-Carleson toy models (corresponding to our article), we can’t use directly Mañé’s hyperbolicity criterion and we have a Cantor set of critical points (we should deal with at the same time). In particular, we replace Mañé’s criterion by its two-dimensional version, namely, the theorem B of E. Pujals and M. Sambarino. This theorem is a fundamental tool used by Pujals and Sambarino in their solution of Palis conjecture for {C^1}-surface diffeomorphisms (stating that Axiom A and homoclinic tangencies are {C^1}-dense among surface diffeomorphisms). In our context, the theorem B of Pujals-Sambarino implies that any maximal invariant set (of Benedicks-Carleson toy models) far away from the critical set is hyperbolic.

Using this fact, it remains to deal with the critical points. Recall that, in the one-dimensional setting, it was very easy to show that the unique critical point is typically recurrent: in fact, since the set of orbits staying away from the critical point form a Cantor set (by Mañé’s criterion), we can make small perturbations of the orbit of the critical point avoids this Cantor set. However, in the two-dimensional setting, we have to ensure that the orbits associated to a Cantor set of critical points avoid the stable manifold of a hyperbolic set. Of course, this is a subtle problem because the perturbations removing some critical points from the stable manifold of a hyperbolic set may create other critical points belonging to this stable manifold. The attentive reader sees that this kind of problem resembles Newhouse phenomena (and stable intersections of Cantor sets) and, actually, this is the case: morally speaking, critical points belonging to stable manifolds of hyperbolic sets corresponds to certain (heteroclinic) tangencies. Thus, we apply Gugu’s theorem (on the non-existence of {C^1}-stable intersections of Cantors sets) to get recurrence of critical points. Finally, using the simple geometry of these models, we combine the recurrence of critical points with an appropriate “flatness” perturbation argument to ensure that the Cantor set of critical points {C^1}-typically falls into the basins of a finite number of sinks.

In any case, this completes our discussion. See you!

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