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 diffeomorphisms of surfaces, namely, Newhouse phenomena (a local mechanism to create robust homoclinic tangencies using -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 -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 -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 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.
Theorem 1 (Jakobson) Axiom A is -dense among the set of unimodal maps of the interval .
Remark 1 Recall that a map of the interval is called unimodal if has exactly one critical point .
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 -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 -density result. We begin with a fundamental criterion of hyperbolicity due to Mañé:
- any periodic orbit of inside with sufficiently high period is a source;
- if every periodic point of inside is a source, then there are some constants and such that for any .
Corollary 3 (Mañé) Let be a endomorphism of the interval and be a compact -invariant set. Assume that all periodic point of are sources and does not contain critical points of . Then, is hyperbolic, i.e., there are constants and such that for all and .
Remark 3 The reader should pay attention to the 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 density of Axiom A among unimodal maps. Let be a unimodal map of the interval . Without loss of generality, we can suppose that:
- is the critical point of and
- is Kupka-Smale (all periodic points are hyperbolic).
For sake of simplicity, we denote by the set of endomorphisms of verifying the three conditions above.
Proof: For each , let be the set of such that either for some or falls into the basin of a sink of . We claim that is open and dense (for every ). Indeed, since is clearly open, our task is reduced to show that is dense.
Given and , consider
Note that is a compact invariant set without critical points and sinks. Since is Kupka-Smale, every periodic point inside must be a source, so that the corollary 3 implies that is hyperbolic, i.e., for some , , we have for all . In particular, is a Cantor set (i.e., a compact set with empty interior). In fact, if the interior of were non-empty, say where is a non-trivial, it would follow that so that the condition of hyperbolicity implies for all . Thus, when , a contradiction with and . On the other hand, since , it follows that .
At this point, we use the following very simple idea: because is a Cantor set, the condition can not be generic (so that it can be destroyed by small perturbation). More precisely, we consider a -small perturbation of supported on such that . We claim that . Indeed, since coincides with outside , we get
Thus, if , we have
Hence, using that by construction, we obtain , that is, is attracted by a sink of . However, since the orbit of under (and consequently ) never touches the interval , one sees that the orbit of sink of attracting belongs to . Therefore, the sink of attracting is also a sink of . In other words, , i.e., , an absurd. This shows that is open and dense for any .
Finally, we complete the proof of the proposition by taking the residual set of . Clearly, any verifies the statement of the proposition. This finishes the proof.
Remark 4 The argument used in the proof of this proposition does not rely on the -topology so that a similar statement holds with respect to the -topology for any .
Up to now, we do not used the -topology in our discussion. However, the next proposition strongly relies on the -topology and it is the main obstruction for an extension of Jakobson’s result to the -topology, for instance.
Proof: Since the 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 such that is very close to and for all .
Denote by . Since , we can made arbitrarily small and we can select such that . For sake of convenience, we take minimal for the property . We have two possibilities:
- (a) is more close to than , i.e., ;
- (b) is more close to than , i.e., .
In the first case (a), we apply a “Closing Lemma” argument: take to be the middle point between and and a small open neighborhood of with for all . See the figure below.
In this situation, we modify inside as follows: denoting by , take -close to so that , has an unique critical point at and outside . It follows that is unimodal with critical point such that is periodic (the fact that the recurrent point becomes periodic justifies the name “Closing Lemma” for this argument). By a -perturbation of , we obtain 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 to be the middle point between and and a small open neighborhood of such that for all with . See the figure below.
In this context, we modify as follows. Take -close to so that is constant on and . Observe that this perturbation can be done (only) in the -topology since we know that is a critical point of and are fairly close to . Next, note that is a periodic point of with period . Moreover, is a super-attracting sink of (because is constant on and thus ). This allows us to make a -perturbation of so that is unimodal with derivative almost zero (but never vanishing) on and possesses a sink whose basin contains . See the figure below.
Consequently, is -close to and . This completes the proof.
Once these two propositions are proved, Jakobson’s theorem follows directly. Indeed, given a endormorphism of the interval , we can assume that (as discussed before). Next, we approximate by a “typical” so that the proposition 4 says that either or .
If , say the critical point falls by iteration into the basin of a sink of ), we see that there exists such that the neighborhood falls into the basin of attraction of the sink . On the other hand, since , we know that is the union of an uniformly expanding hyperbolic set and a finite number of sinks. In particular, we get that is an Axiom A endomorphism arbitrarily -close to .
If (i.e., the critical point is recurrent), we apply the proposition 5 to get an endomorphism arbitrarily -close to so that . Therefore, the discussion of the previous paragraph gives us some Axiom A endomorphism arbitrarily -close to .
Thus, in any case, can be -approximated by an Axiom A endomorphism. This completes the proof of Jakobson’s theorem.
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 -surface diffeomorphisms (stating that Axiom A and homoclinic tangencies are -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 -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 -typically falls into the basins of a finite number of sinks.
In any case, this completes our discussion. See you!