In the previous post of this series, we considered a horseshoe of a , , diffeomorphism of a compact surface with some periodic point involved in a quadratic homoclinic tangency. Also, for sake of simplicity, we assumed that there are two neighborhoods of the horseshoe and of the homoclinic orbit of such that
and we organized the parameter space given by a sufficiently small -neighborhood of into depending on the number of intersections near of the invariant manifolds of , that is,
Finally, we noticed that
- for any , and
- for any ,
so the homoclinic bifurcations (new dynamical phenomena) can occur only on .
Here, it may be tempting to try to understand the local dynamics of all . However, after the seminal works of Sheldon Newhouse, one knows that it is not reasonable to try to control for all because of a mechanism nowadays called Newhouse phenomena. In fact, the Newhouse phenomena were already mentioned a few years ago in this blog, but the discussion there was somewhat superficial. So, we will use today’s post to explain (below the fold) Newhouse mechanism with a little bit more of details. Again, the basic reference for this post is Palis-Takens’ book.
It is clear from Figure 0 above that by unfolding a quadratic tangency to get an element , we end up with some horseshoes near the region : indeed, the presence of transverse homoclinic orbits in of implies the existence of some horseshoes by the discussion of the previous subsection. In particular, this naive argument seems to ensure that is always hyperbolic. However, S. Newhouse noticed that by unfolding the quadratic tangency associated to we may create other tangencies nearby, that is, we may “accidentally” lose the hyperbolicity just created in view of the incompatibility between tangencies and hyperbolicity.
In fact, as we’re going to explain in a moment, this “accidental” formation of new tangencies happens especially when the horseshoe containing is thick (“fat”). In this direction, the first step (a classical one in Dynamics) is to “reduce” the detection of tangencies for diffeomorphisms of -dimensional manifolds to the -dimensional problem of understanding the intersection of two Cantor sets in .
1. Persistence of tangencies and intersections of Cantor sets
Starting from , we consider an extension , resp., , of the stable, resp. unstable, laminations of to the neighborhood of . Such an extension exists by the results of Welington de Melo and it heavily depends on the fact that is a diffeomorphism of a –dimensional manifold.
From the fact that is a homoclinic quadratic tangency associated to , one deduces that the foliations and meet tangencially along a curve called line of tangencies. Using as an auxiliary curve, we can study the formation of tangencies for (i.e., after unfolding the quadratic tangency of ) as follows. One considers the (local) Cantor sets and , and, by using the holonomy of the stable, resp. unstable, foliations , resp. , one can diffeomorphically map and into Cantor sets and by sending , resp. , to , resp. . Note that, by definition, the intersection of and corresponds to all tangencies between the stable and unstable laminations of the horseshoe near , that is, by our assumptions, . Pictorially, the discussion of this paragraph can be resumed by the following figure:
Now, let us perturb this picture by unfolding the tangency to get some . It is possible to show that this picture admits a natural continuation because the relevant dynamical objects have continuations (we’ll comment more on this in Section 2, but for now let’s postpone this “continuity” discussion), that is, one can extend the stable and unstable laminations of to stable and unstable foliations and of a neighborhood (“close” to and ), and one can use them to define a line of tangencies (“close” to ) containing two Cantor sets and (“close” to and ) defined as the images of the Cantor sets and via the stable and unstable holonomies and . Also, the intersection of the Cantor sets and still accounts for all tangencies between the stable and unstable laminations of the horseshoe . In resume, we get the following local dynamical picture for :
In particular, the problem of persistent tangencies for all , i.e., the issue that the stable and unstable laminations of meet tangentially at some point in for all , is reduced to the question of studying sufficient conditions for a non-trivial intersection between the Cantor sets and of the line .
2. Intersections of thick Cantor sets and Newhouse gap lemma
By thinking of as a subset of the real line , our current objective is clearly equivalent to produce sufficient conditions to ensure that two Cantor sets in have non-trivial intersection. Keeping this goal in mind, S. Newhouse introduced a notion of thickness of a Cantor set :
Definition 1 Let be a Cantor set. A gap of is a connected component of , and a bounded gap of is a bounded connected component of . Given a bounded gap of and , the bridge of at is the maximal interval such that and contains no gap of with length greater than or equal to the lenght of . In this language, the thickness of at is and the thickness of is
The thickness is a nice concept for our purposes because of the following fundamental result nowadays called gap lemma:
Lemma 2 (Newhouse gap lemma) Let and be two thick Cantor sets of in the sense that . Then, one of the following possibilities occur:
- is contained in a gap of (i.e., a connected component of ),
- is contained in a gap of ,
Proof: Let and be two Cantor sets such that
- (a) is not contained in a gap of and
- (b) is not contained in a gap of .
We wish to show that when . We will proceed by contradiction, that is, let us suppose also that .
We say that a pair of bounded gaps of is linked whenever . Pictorially, a linked pair of gaps looks like this:
We claim that there exists a linked pair of bounded gaps when the items (a) and (b) are satisfied and . Indeed, let , resp. , be the closed interval obtained as the convex hull of , resp. . By the items (a) and (b), is not contained in the (unbounded) gaps and of , and is not contained in the (unbounded) gaps and of . Also, from the disjointness assumption and the fact that and are the convex hulls of the Cantor sets and , we have that and . Hence, we have that either , or the intervals and linked in the same sense as before, that is, up to exchanging the roles of and , one has (i.e., one has the same picture as in Figure 2 after replacing and by and resp.). In any event, up to exchanging the roles of and , we may assume that the point belongs to .
Since is the convex hull of the Cantor set , . Hence, by the disjointness assumption , , where is a bounded gap of . Now, we consider the point . Since is a gap of , we have that , so that, by the disjointness assumption , we can find a bounded gap of such that . Note that this proves our claim because is a linked pair of gaps of : indeed, this follows from the facts that , and . The following figure illustrates the argument proving this claim.
Once we dispose a linked pair of gaps of , we will use the hypothesis to reach a contradiction with the disjointness assumption via the following algorithm. Starting with a linked pair , , of bounded gaps of as shown in Figure 2, denote by , resp. , the bridge of , resp. , at the rightmost, resp. leftmost, point in , resp. . By the definition of the thickness of a Cantor set, the condition implies that
Therefore, we have that either or . So, either the rightmost point of belongs to or the leftmost point of belongs to (see the figure below).
Up to exchanging the roles of and , we may assume that the first possibility occurs. In this situation, since and , we have that belongs to a gap of . Furthermore, since belongs to the bridge , we have, by definition, that . Moreover, is a linked pair of bounded gaps of .
In resume, starting with a linked pair of bounded gaps of , a pair of disjoint Cantor sets with , we can construct or a linked pair of bounded gaps of such that either or .
By iterating the algorithm described above, we can use our initial linked pair of gaps to produce a sequence of linked pairs of bounded gaps of such that either or . Since the sum of the lengths of bounded gaps of , resp. , is finite (, resp. ), we have that either or as . In particular, either
At this point, we have a contradiction because is linked, and thus
- either accumulated by the sequence of points and hence ,
- or accumulated by the sequence of points and hence .
In any event, this completes the proof of Newhouse gap lemma.
Remark 1 A practical way of using Newhouse gap lemma by looking at the relative position of two Cantor sets is the following. We say that two Cantor sets are linked whenever their convex hulls are linked in the sense that but neither nor . Then, by Newhouse’s gap lemma, two linked Cantor sets with must intersect non-trivially because, as the reader can easily check, the assumption that and are linked prohibits a gap of to contain and vice-versa.
Of course, as the reader can imagine, the gap lemma put us in position to come back to the discussion of persistence of tangencies for . Indeed, the statement of the gap lemma hints that one has persistence of tangencies for all as soon as for the initial dynamics . Actually, this fact was shown to be true by S. Newhouse, but this is not an immediate consequence of his gap lemma because we need to know that the Cantor sets and are thick for all and we have only that and , , are thick!
At this point, the idea (already mentioned above) is to play with “continuity” of dynamical objects: intuitively, and must be thick because they are “close” to the thick Cantor sets and . However, the formal implementation of this idea is rather technical and we’ll content ourselves with a mere description of the crucial points of the argument.
3. Continuity of thickness and Newhouse phenomena
Firstly, one has to explain what does it mean for to be “close” to : for our purposes, we’ll say that is close to whenever the map is -close to the map . Here, is our preferred (small) neighborhood of the horseshoe of and , resp. , is the leaf of , resp. passing through .
As it is shown in Theorem 8 of Appendix 1 of Palis-Takens book, is close to when is -close to for . Using this result, one can show that the line of tangency is -close to , and the projections and are -close to and . An immediate consequence of this is: one can nicely identify both and with the interval in such a way that is close to in the Hausdorff topology (we say that a compact subset is -close to a compact subset in the Hausdorff topology when for each there exists with , and for each there exists with ). However, this is not very useful because it is not true in general that is close to when and are close in Hausdorff topology. Here, the idea to overcome this difficulty relies on the observation that and belong to a special class of Cantor sets known as regular (dynamically defined) Cantor sets.
More precisely, we say that a Cantor set is –regular, , if there are disjoint compact intervals and an (uniformly) expanding function (i.e., for any ) from the disjoint union to its convex hull such that:
- , that is, is defined by the dynamics , and
- the collection is a Markov partition: for any , the interval is the convex hull of the union of some of the intervals and for some large .
Regular (dynamically defined) Cantor sets are very common in nature. For example, the classical ternary Cantor set is a regular Cantor set. Indeed, it is not hard to see that where is the (piecewise affine) expanding function defined by
Another example are the Cantor sets and : as it is shown in Chapter 4 of Palis-Takens book, they are -regular Cantor sets whenever and are -diffeomorphisms.
The class of -regular Cantor sets admit a natural –topology: we say that two -regular Cantor sets and are –close whenever the extremal points of the associated intervals and are close and the expanding functions and are -close. For example, the -regular Cantor sets and are -close when and are -close.
These definitions are well-adapted to the study of homoclinic tangencies because of the following fundamental fact:
See Chapter 4 of Palis-Takens book for more discussion on this proposition.
Hence, if is a -diffeomorphism, , and , then for all . Thus, since is -close to , and and are close to and , it is possible to compare and via a -diffeomorphism whose derivative is -close to the identity. In particular, since -diffeomorphisms with a derivative -close to the identity don’t change in a drastic way the thickness, one has that is close , so is close to . Hence, we conclude that for all . Moreover, for each , the Cantor sets and of the line are linked (as one can see from Figure 1). Thus, by Newhouse’s gap lemma (or more precisely Remark 1), we obtain that for all .
In other words, we have just outlined the proof of the following result about persistance of tangencies:
Then, for all , the stable and unstable laminations of the horseshoe intersect tangentially at some point in .
Once we dispose of this theorem on persistence of tangencies in our toolbox, we’re ready to discuss the Newhouse phenomena. Again, we start with a -diffeomorphism with and we now assume that:
- the periodic point is dissipative, i.e., where is the period of , and
For sake of concreteness, let’s suppose that . Note that this implies for all . Denote by the eigenvalues of , so that . Now, given , we know by Theorem 4that the stable and unstable laminations of meet tangentially at some point in . Since the stable and unstable manifolds of the periodic point are dense in the stable and unstable laminations of (cf. Chapter 2 of Palis-Takens book), one can apply arbitrarily small perturbations to so that there is no loss of generality in assuming that and meet tangentially at some point in the region . Starting from this quadratic tangency, one gets the following picture for diffeomorphisms close to :
As it is shown in Chapter 3 of Palis-Takens book (see pages 46–52), one can carefully choose parameters () such that
- as and
- the map can be renormalized in such a way that the renormalizations of -converge to the endomorphism .
Remark 2 In principle, the parameters must vary in some infinite-dimensional manifold in order to parametrize a neighborhood of , but for sake of simplicity of the exposition, we will think of this parameter as a real number “measuring” the distance between the line and the “tip” of the parabola as indicated in Figure 3.
Remark 3 “ can be renormalized” means that one can perform an adequate –dependent change of coordinates on to get a new dynamics .
Note that the diffeomorphisms are converging to an endomorphism and this may seem strange at first sight. However, this is natural in view of the assumption that the periodic point is dissipative: in fact, the area-contraction condition says that become strongly area-contracting as and consequently converges (after appropriate “scaling”/renormalization) to a curve and converges to an endomorphism of this curve as .
Next, we observe that the endomorphism has an attracting fixed point at . Therefore, by convergence of towards this endomorphism, we conclude that has an attracting fixed point in for all sufficiently large. In other words, has a sink (attracting periodic point) in the region for all sufficiently small.
This last statement can be reformulated as follows. For each , denote by . Note that is open for all (because any sink is persistence under small perturbations of the dynamics). Moreover, since was “arbitrary” in the previous argument, we also have that is dense in .
At this stage, the idea of S. Newhouse is to iterate this argument to show that the set
of diffeomorphisms of with infinitely many sinks is residual in Baire sense(and, in particular, is dense in ). Since is open in for all and is dense in , it suffices to prove that is dense in for all to conclude that is residual.
In this direction, one starts with with periodic sinks . By Theorem 4, we know that the stable and unstable laminations of meet tangentially somewhere in . Since , resp. , is dense in the stable, resp. unstable, lamination of , we can assume (up to performing an arbitrarily small perturbation on ) that and meet tangentially at some point and has periodic sinks. Next, we select a small neighborhood of such that none of the periodic sinks passes through , i.e., for each . By repeating the “renormalization” arguments above (with replaced by ), one can produce a sequence of diffeomorphisms converging to as such that has a sink passing through . Because the sinks don’t pass through for all sufficiently large, this means that is a new sink of , that is, we obtain that for all sufficiently large. Since as , we conclude that is dense in .
In resume, we gave a sketch of the proof of the following result:
- the periodic point is dissipative, say where is the period of , and
Then, the subset of diffeomorphisms with infinitely many sinks is residual.
This result of coexistence of infinitely many sinks for a residual (and, hence, dense) subset of diffeomorphisms of is the so-called Newhouse phenomena. This theorem is very important because it says that for a topologically big (residual) set of diffeomorphisms the dynamics is so complicated that there are infinitely many attractors. Thus, if we pick at random a point of , it is very hard to decide (from the computational point of view for instance) the future of the orbit of because it can be attracted by anyone of the infinitely many sinks.
In other words, the Newhouse phenomena says that it is not reasonable to try to understand the local dynamics of all . At this point, since we know that it is too naive to try to dynamically describe all , we can ask:
What about the local dynamical behavior of most (whatever the word “most” means)?
This question will occupy the next posts. For now, we close today’s discussion with two comments:
Remark 4 Actually, S. Newhouse proved in this paper here (see also Chapter 6 of Palis-Takens book) that one can remove the second assumption (on thicknesses) in the statement of Theorem 5: indeed, starting with any dissipative area-contracting hyperbolic periodic point (of saddle-type) of a -diffeomorphism having some point of tangency between and , S. Newhouse can construct (by further analyzing the renormalization argument described above) open sets arbitrarily close to such that the subset of diffeomorphisms of with infinitely many sinks is residual in .
Remark 5 The attentive reader certainly noticed that we insisted that Newhouse phenomena (Theorem 5) concerns -diffeomorphisms. In fact, this regularity assumption is crucial to get the continuity of the thickness of regular Cantor sets in Proposition 3. Indeed, the proof of this proposition in Chapter 4 of Palis-Takens book (or alternatively in this post here) strongly relies on the so-called bounded distortion property saying that the shape of gaps and bridges of a -regular Cantor set is “essentially constant in all scales”. Of course, the continuity of thickness is one of the central mechanisms for Newhouse phenomena (as it ensures that the Cantor sets and intersect for all ) and the reader may be curious whether the Newhouse phenomena occurs for -diffeomorphisms. As a matter of fact, it is known that the thickness of -regular Cantor sets is not continuous, so that the Newhouse gap lemma can’t be applied in the -context. Of course, it could be that -regular Cantor sets intersect often/stably, thus giving some hope for an analog of Newhouse’s thickness mechanism to survive in the -setting. However, this possibility was recently dismissed by C. (Gugu) Moreira (see these posts here) and this “absence of Newhouse mechanism” was used by C. Moreira, E. Pujals and the author (see this post here) to check that among certain families of dynamical systems it is possible to get a sort of Newhouse phenomena in the -setting but still the -generic element of the family has finitely many sinks.