Posted by: matheuscmss | August 30, 2012

Homoclinic/heteroclinic bifurcations: Newhouse phenomena

In the previous post of this series, we considered a horseshoe {K} of a {C^k}, {k\geq 2}, diffeomorphism {f:M\rightarrow M} of a compact surface {M} with some periodic point {p\in K} involved in a quadratic homoclinic tangency. Also, for sake of simplicity, we assumed that there are two neighborhoods {U} of the horseshoe {K} and {V} of the homoclinic orbit {\mathcal{O}(q):=\{f^n(q):n\in\mathbb{Z}\}} of {q} such that

\displaystyle \bigcap\limits_{n\in\mathbb{Z}} f^n(U\cup V)=K\cup \mathcal{O}(q).

and we organized the parameter space {\mathcal{U}} given by a sufficiently small {C^k}-neighborhood of {f} into {\mathcal{U}=\mathcal{U}_-\cup\mathcal{U}_0\cup\mathcal{U}_+} depending on the number of intersections near {q} of the invariant manifolds of {p}, that is,

Figure 0. Organization of the parameter space {\mathcal{U}}.

Finally, we noticed that

  • {\Lambda_g=K_g} for any {g\in\mathcal{U}_-}, and
  • {\Lambda_g=K_g\cup \mathcal{O}(q_g)} for any {g\in\mathcal{U}_0},

so the homoclinic bifurcations (new dynamical phenomena) can occur only on {\mathcal{U}_+}.

Here, it may be tempting to try to understand the local dynamics of all {g\in\mathcal{U}_+}. However, after the seminal works of Sheldon Newhouse, one knows that it is not reasonable to try to control {\Lambda_g} for all {g\in\mathcal{U}_+} 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 {g\in\mathcal{U}_+}, we end up with some horseshoes near the region {V}: indeed, the presence of transverse homoclinic orbits in {V} of {g\in\mathcal{U}_+} implies the existence of some horseshoes by the discussion of the previous subsection. In particular, this naive argument seems to ensure that {\Lambda_g} is always hyperbolic. However, S. Newhouse noticed that by unfolding the quadratic tangency associated to {p} 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 {K} containing {p} is thick (“fat”). In this direction, the first step (a classical one in Dynamics) is to “reduce” the detection of tangencies for diffeomorphisms of {2}-dimensional manifolds to the {1}-dimensional problem of understanding the intersection of two Cantor sets in {\mathbb{R}}.

1. Persistence of tangencies and intersections of Cantor sets

Starting from {f\in\mathcal{U}_0}, we consider an extension {\mathcal{F}^s(f)}, resp., {\mathcal{F}^u(f)}, of the stable, resp. unstable, laminations of {K} to the neighborhood {U} of {K}. Such an extension exists by the results of Welington de Melo and it heavily depends on the fact that {f} is a diffeomorphism of a {2}dimensional manifold.

From the fact that {q} is a homoclinic quadratic tangency associated to {p}, one deduces that the foliations {\mathcal{F}^s} and {\mathcal{F}^u} meet tangencially along a curve {\ell=\ell(f)} called line of tangencies. Using {\ell(f)} as an auxiliary curve, we can study the formation of tangencies for {g\in\mathcal{U}_+} (i.e., after unfolding the quadratic tangency of {f}) as follows. One considers the (local) Cantor sets {W_{loc}^s(p)\cap K} and {W_{loc}^u(p)\cap K}, and, by using the holonomy of the stable, resp. unstable, foliations {\mathcal{F}^s(f)}, resp. {\mathcal{F}^u(f)}, one can diffeomorphically map {W_{loc}^s(p)\cap K} and {W_{loc}^u(p)\cap K} into Cantor sets {K^s\subset\ell} and {K^u\subset \ell} by sending {x\in W_{loc}^s(p)\cap K}, resp. {x\in W_{loc}^s(p)\cap K}, to {\pi_f^s(x)=y\in W^s(x)\cap\ell}, resp. {\pi_f^u(x)=y\in W^u(p)\cap\ell}. Note that, by definition, the intersection of {K^s} and {K^u} corresponds to all tangencies between the stable and unstable laminations of the horseshoe {K} near {V}, that is, by our assumptions, {K^s\cap K^u=\{q\}}. Pictorially, the discussion of this paragraph can be resumed by the following figure:

The line of tangencies {\ell} and the Cantor sets {K^s} and {K^u} for {f\in\mathcal{U}_0}: the crosses are points in {K^s} and the dots are points in {K^u}.

Now, let us perturb this picture by unfolding the tangency to get some {g\in\mathcal{U}_+}. 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 {K_g} to stable and unstable foliations {\mathcal{F}^s(g)} and {\mathcal{F}^u(g)} of a neighborhood {U} (“close” to {\mathcal{F}^s(f)} and {\mathcal{F}^u(f)}), and one can use them to define a line of tangencies {\ell(g)} (“close” to {\ell(f)}) containing two Cantor sets {K^s(g)} and {K^u(g)} (“close” to {K^s} and {K^u}) defined as the images of the Cantor sets {W_{loc}^s(p_g)\cap K_g} and {W_{loc}^u(p_g)\cap K_g} via the stable and unstable holonomies {\pi^s_g} and {\pi^u_g}. Also, the intersection {K^s(g)\cap K^u(g)} of the Cantor sets {K^s(g)} and {K^u(g)} still accounts for all tangencies between the stable and unstable laminations of the horseshoe {K_g}. In resume, we get the following local dynamical picture for {g\in\mathcal{U}_+}:

Figure 1. The line of tangencies {\ell(g)} and the Cantor sets {K^s(g)} and {K^u(g)} for {g\in\mathcal{U}_+}: the crosses are points in {K^s(g)} and the dots are points in {K^u(g)}.

In particular, the problem of persistent tangencies for all {g\in\mathcal{U}_+}, i.e., the issue that the stable and unstable laminations of {K_g} meet tangentially at some point in {V} for all {g\in\mathcal{U}_+}, is reduced to the question of studying sufficient conditions for a non-trivial intersection {K^s(g)\cap K^u(g)\neq\emptyset} between the Cantor sets {K^s(g)} and {K^u(g)} of the line {\ell(g)}.

2. Intersections of thick Cantor sets and Newhouse gap lemma

By thinking of {\ell(g)} as a subset of the real line {\mathbb{R}}, our current objective is clearly equivalent to produce sufficient conditions to ensure that two Cantor sets in {\mathbb{R}} have non-trivial intersection. Keeping this goal in mind, S. Newhouse introduced a notion of thickness {\tau(C)} of a Cantor set {K\subset\mathbb{R}}:

Definition 1 Let {K\subset\mathbb{R}} be a Cantor set. A gap {U} of {K} is a connected component of {\mathbb{R}-K}, and a bounded gap {U} of {K} is a bounded connected component of {\mathbb{R}-K}. Given {U} a bounded gap of {K} and {u\in\partial U}, the bridge {C} of {K} at {u} is the maximal interval {C\subset\mathbb{R}} such that {u\in\partial C} and {C} contains no gap {U'} of {K} with length {|U'|} greater than or equal to the lenght {|U|} of {U}. In this language, the thickness of {K} at {u} is {\tau(K,u):=|C|/|U|} and the thickness{\tau(K)} of {K} is

\displaystyle \tau(K)=\inf\limits_{u}\tau(K,u).

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 {K} and {K'} be two thick Cantor sets of {\mathbb{R}} in the sense that {\tau(K)\cdot\tau(K')>1}. Then, one of the following possibilities occur:

  • {K'} is contained in a gap of {K} (i.e., a connected component of {\mathbb{R}-K}),
  • {K} is contained in a gap of {K'},
  • {K\cap K'\neq\emptyset}.

Proof: Let {K} and {K'} be two Cantor sets such that

  • (a) {K'} is not contained in a gap of {K} and
  • (b) {K} is not contained in a gap of {K'}.

We wish to show that {K\cap K'\neq\emptyset} when {\tau(K)\cdot\tau(K')>1}. We will proceed by contradiction, that is, let us suppose also that {K\cap K'=\emptyset}.

We say that a pair {(U,U')} of bounded gaps of {(K, K')} is linked whenever {\#(U\cap\partial U')=1=\#(U\cap\partial U')}. Pictorially, a linked pair {(U, U')} of gaps looks like this:

Figure 2. A linked pair of gaps.

We claim that there exists a linked pair {(U_1,U_2)} of bounded gaps when the items (a) and (b) are satisfied and {K\cap K'=\emptyset}. Indeed, let {I=[x,y]}, resp. {I'=[x',y']}, be the closed interval obtained as the convex hull of {K}, resp. {K'}. By the items (a) and (b), {K} is not contained in the (unbounded) gaps {(-\infty, x')} and {(y',\infty)} of {K'}, and {K'} is not contained in the (unbounded) gaps {(-\infty, x)} and {(y,\infty)} of {K}. Also, from the disjointness assumption {K\cap K'=\emptyset} and the fact that {I} and {I'} are the convex hulls of the Cantor sets {K} and {K'}, we have that {K\ni x\neq x'\in K'} and {K\ni y\neq y'\in K'}. Hence, we have that either {I\subset I'}{I'\subset I} or the intervals {I} and {I'}  linked in the same sense as before, that is, up to exchanging the roles of {K} and {K'}, one has {x<x'<y<y'} (i.e., one has the same picture as in Figure 2 after replacing {U} and {U'} by {I} and {I'} resp.). In any event, up to exchanging the roles of {I} and {I'}, we may assume that the point {y\in \partial I} belongs to {I'}.

Since {I} is the convex hull of the Cantor set {K}, {y\in K}. Hence, by the disjointness assumption {K\cap K'=\emptyset}, {y\in U'}, where {U'\subset I'} is a bounded gap of {K'}. Now, we consider the point {z\in I\cap \partial U'}. Since {U'} is a gap of {K'}, we have that {z\in K'}, so that, by the disjointness assumption {K\cap K'=\emptyset}, we can find {U\subset I} a bounded gap of {K} such that {z\in U}. Note that this proves our claim because {(U, U')} is a linked pair of gaps of {(K, K')}: indeed, this follows from the facts that {U\subset I=[x,y]}, {z\in U\cap\partial U'} and {y\in U'}. The following figure illustrates the argument proving this claim.

Construction of a linked pair of gaps.

Once we dispose a linked pair {(U,U'):=(U_0,U_0')} of gaps of {(K,K')}, we will use the hypothesis {\tau(K)\cdot\tau(K')>1} to reach a contradiction with the disjointness assumption {K\cap K'=\emptyset} via the following algorithm. Starting with a linked pair {(U_i,U_i')}, {i\in\mathbb{N}}, of bounded gaps of {(K,K')} as shown in Figure 2, denote by {C_i}, resp. {C_i'}, the bridge of {K}, resp. {K'}, at the rightmost, resp. leftmost, point in {\partial U_i}, resp. {\partial U_i'}. By the definition of the thickness of a Cantor set, the condition {\tau(K)\cdot \tau(K')>1} implies that

\displaystyle \frac{|C_i|}{|U_i|}\cdot\frac{|C_i'|}{|U_i'|}>1

Therefore, we have that either {|C_i|>|U_i'|} or {|C_i'|>|U_i|}. So, either the rightmost point {u_i'} of {\partial U_i'} belongs to {C_i} or the leftmost point {u_i} of {\partial U_i} belongs to {C_i'} (see the figure below).

Partial description of the algorithm for the construction of a sequence of linked pairs of gaps. Here, {|C_i|>|U_i'|} and {|C_i'|>|U_i|}.

Up to exchanging the roles of {K} and {K'}, we may assume that the first possibility occurs. In this situation, since {u_i'\in \partial U_i'\subset K'} and {K\cap K'=\emptyset}, we have that {u_i'} belongs to a gap {U_{i+1}} of {K}. Furthermore, since {u_i'} belongs to the bridge {C_i}, we have, by definition, that {|U_{i+1}|<|U_i|}. Moreover, {(U_{i+1}, U_{i+1}'):=(U_{i+1}, U_i')} is a linked pair of bounded gaps of {(K, K')}.

In resume, starting with a linked pair {(U_i, U_i')} of bounded gaps of {(K, K')}, a pair of disjoint Cantor sets with {\tau(K)\cdot \tau(K')>1}, we can construct {(U_{i+1}, U_{i+1}'):=(U_{i+1}, U_i')} or {(U_i, U_{i+1}')} a linked pair of bounded gaps of {(K, K')} such that either {|U_{i+1}|<|U_i|} or {|U_{i+1}'|<|U_i'|}.

By iterating the algorithm described above, we can use our initial linked pair {(U_0,U_0')} of gaps to produce a sequence {(U_i, U_i')} of linked pairs of bounded gaps of {(K,K')} such that either {|U_{i+1}|<|U_i|} or {|U_{i+1}'|<|U_i'|}. Since the sum of the lengths of bounded gaps of {K}, resp. {K'}, is finite ({\leq|y-x|=|I|<\infty}, resp. {\leq|y'-x'|=|I'|<\infty}), we have that either {|U_i|\rightarrow 0} or {|U_i'|\rightarrow 0} as {i\rightarrow\infty}. In particular, either

\displaystyle \bigcap\limits_{i\in\mathbb{N}} U_i =\{p\} \quad \textrm{with } p\in K

or

\displaystyle \bigcap\limits_{i\in\mathbb{N}} U_i' =\{p'\} \quad \textrm{with } p'\in K'.

At this point, we have a contradiction because {(U_i, U_i')} is linked, and thus

  • either {p\in K} accumulated by the sequence of points {U_i\cap\partial U_i'\in K'} and hence {p\in K\cap K'},
  • or {p'\in K'} accumulated by the sequence of points {U_i'\cap\partial U_i\in K} and hence {p\in K'\cap K}.

In any event, this completes the proof of Newhouse gap lemma. \Box

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 {K,K'\subset\mathbb{R}} are linked whenever their convex hulls {I, I'} are linked in the sense that {I\cap I'\neq\emptyset} but neither {I\not\subset I'} nor {I'\not\subset I}. Then, by Newhouse’s gap lemma, two linked Cantor sets {K,K'\subset\mathbb{R}} with {\tau(K)\cdot\tau(K')>1} must intersect non-trivially because, as the reader can easily check, the assumption that {K} and {K'} are linked prohibits a gap of {K'} to contain {K} 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 {g\in\mathcal{U}_+}. Indeed, the statement of the gap lemma hints that one has persistence of tangencies for all {g\in\mathcal{U}_+} as soon as {\tau(K^s)\cdot \tau(K^u)>1} for the initial dynamics {f\in\mathcal{U}_0}. 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 {K^s(g)} and {K^u(g)} are thick for all {g\in\mathcal{U}_+} and we have only that {K^s=K^s(f)} and {K^u=K^u(f)}, {f\in\mathcal{U}_0}, are thick!

At this point, the idea (already mentioned above) is to play with “continuity” of dynamical objects: intuitively, {K^s(g)} and {K^u(g)} must be thick because they are “close” to the thick Cantor sets {K^s} and {K^u}. 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 {\mathcal{F}^s(g)} to be “close” to {\mathcal{F}^s(f)}: for our purposes, we’ll say that {\mathcal{F}^s(g)} is close to {\mathcal{F}^s(f)} whenever the map {U\ni x\mapsto T_x\mathcal{F}^s(f)(x)} is {C^1}-close to the map {U\ni x\mapsto T_x\mathcal{F}^s(g)(x)}. Here, {U} is our preferred (small) neighborhood of the horseshoe {K} of {f} and {\mathcal{F}^s(f)(x)}, resp. {\mathcal{F}^s(g)(x)}, is the leaf of {\mathcal{F}^s(f)}, resp. {\mathcal{F}^s(g)} passing through {x}.

As it is shown in Theorem 8 of Appendix 1 of Palis-Takens book, {\mathcal{F}^s(g)} is close to {\mathcal{F}^s(f)} when {g} is {C^k}-close to {f} for {k\geq 2}. Using this result, one can show that the line of tangency {\ell(g)} is {C^1}-close to {\ell(f)}, and the projections {\pi^s_g} and {\pi^u_g} are {C^1}-close to {\pi^s_f} and {\pi^u_f}. An immediate consequence of this is: one can nicely identify both {\ell(g)} and {\ell} with the interval {I=[0,1]\subset\mathbb{R}} in such a way that {K^s(g)} is close to {K^s (=K^s(f))} in the Hausdorff topology (we say that a compact subset {A\subset\mathbb{R}} is {\delta}-close to a compact subset {B\subset\mathbb{R}} in the Hausdorff topology when for each {y\in B} there exists {x\in A} with {|y-x|<\delta}, and for each {w\in A} there exists {z\in B} with {|w-z|<\delta}). However, this is not very useful because it is not true in general that {\tau(K^s(g))} is close to {\tau(K^s)} when {K^s(g)} and {K^s} are close in Hausdorff topology. Here, the idea to overcome this difficulty relies on the observation that {K^s(g)} and {K^s} belong to a special class of Cantor sets known as regular (dynamically defined) Cantor sets.

More precisely, we say that a Cantor set {K\subset\mathbb{R}} is {C^r}regular, {r\geq 1}, if there are disjoint compact intervals {I_1,\dots,I_l\subset\mathbb{R}} and an (uniformly) expanding {C^r} function {\psi:I_1\cup\dots\cup I_l\rightarrow I} (i.e., {|\psi'(x)|>1} for any {x}) from the disjoint union {I_1\cup\dots\cup I_l} to its convex hull {I} such that:

  • {K=\bigcap\limits_{n\in\mathbb{N}}\psi^{-n}(I)}, that is, {K} is defined by the dynamics {\psi}, and
  • the collection {\{I_1,\dots,I_l\}} is a Markov partition: for any {1\leq j\leq l}, the interval {\psi(I_j)} is the convex hull of the union of some of the intervals {I_i} and {\psi^{n(j)}(I_j)\supset I_1\cup\dots\cup I_l} for some large {n(j)\in\mathbb{N}}.

Regular (dynamically defined) Cantor sets are very common in nature. For example, the classical ternary Cantor set {K_0} is a regular Cantor set. Indeed, it is not hard to see that {K_0=\bigcap\limits_{n\in\mathbb{N}}\psi^{-n}([0,1])} where {\psi:[0,1/3]\cup [2/3,1]\rightarrow [0,1]} is the (piecewise affine) expanding function defined by

\displaystyle \psi(x)=\left\{\begin{array}{ll}3x & \textrm{ if } x\in [0,1/3] \\ 3x-2 & \textrm{ if } x\in [2/3,1] \end{array}\right.

Another example are the Cantor sets {W^s(p)\cap K} and {W^s(p_g)\cap K_g}: as it is shown in Chapter 4 of Palis-Takens book, they are {C^k}-regular Cantor sets whenever {f} and {g} are {C^k}-diffeomorphisms.

The class of {C^r}-regular Cantor sets admit a natural {C^r}topology: we say that two {C^r}-regular Cantor sets {K} and {\widetilde{K}} are {C^r}close whenever the extremal points of the associated intervals {I_1,\dots, I_l} and {\widetilde{I_1},\dots,\widetilde{I_l}} are close and the expanding functions {\psi} and {\widetilde{\psi}} are {C^r}-close. For example, the {C^k}-regular Cantor sets {W^s(p)\cap K} and {W^s(p_g)\cap K_g} are {C^k}-close when {f} and {g} are {C^k}-close.

These definitions are well-adapted to the study of homoclinic tangencies because of the following fundamental fact:

Proposition 3 The thickness of {C^k}-regular Cantor sets vary continuously in the {C^k}-topology for {k\geq 2}.

See Chapter 4 of Palis-Takens book for more discussion on this proposition.

Hence, if {f\in\mathcal{U}_0} is a {C^k}-diffeomorphism, {k\geq 2}, and {\tau(W^s_{loc}(p)\cap K)\cdot\tau(W^u_{loc}(p)\cap K)>1}, then {\tau(W^s_{loc}(p_g)\cap K_g)\cdot\tau(W^u_{loc}(p_g)\cap K_g)>1} for all {g\in\mathcal{U}_+}. Thus, since {\ell(g)} is {C^1}-close to {\ell(f)}, and {\pi^s(g)} and {\pi^u(g)} are {C^1} close to {\pi^s(f)} and {\pi^u(f)}, it is possible to compare {W^s(p_g)\cap K_g} and {K^s(g)} via a {C^1}-diffeomorphism whose derivative is {C^0}-close to the identity. In particular, since {C^1}-diffeomorphisms with a derivative {C^0}-close to the identity don’t change in a drastic way the thickness, one has that {\tau(K^s(g))} is close {\tau(W^s(p_g)\cap K_g)}, so {\tau(K^s(g))\cdot \tau(K^u(g))} is close to {\tau(W^s_{loc}(p_g)\cap K_g)\cdot\tau(W^u_{loc}(p_g)\cap K_g)>1}. Hence, we conclude that {\tau(K^s(g))\cdot \tau(K^u(g))>1} for all {g\in\mathcal{U_+}}. Moreover, for each {g\in\mathcal{U}_+}, the Cantor sets {K^s(g)} and {K^u(g)} of the line {\ell(g)} are linked (as one can see from Figure 1). Thus, by Newhouse’s gap lemma (or more precisely Remark 1), we obtain that {K^s(g)\cap K^u(g)\neq\emptyset} for all {g\in\mathcal{U}_+}.

In other words, we have just outlined the proof of the following result about persistance of tangencies:

Theorem 4 Let {f\in\mathcal{U}_0} be a {C^k}-diffeomorphism, {k\geq 2}, and suppose that

\displaystyle \tau(W^s_{loc}(p)\cap K)\cdot\tau(W^u_{loc}(p)\cap K)>1.

Then, for all {g\in\mathcal{U}_+}, the stable and unstable laminations of the horseshoe {K_g} intersect tangentially at some point in {V}.

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 {C^k}-diffeomorphism {f\in\mathcal{U}_0} with {k\geq 2} and we now assume that:

  • the periodic point {p} is dissipative, i.e., {|\det df^n(p)|\neq 1} where {n} is the period of {p}, and
  • {\tau(W^s_{loc}(p)\cap K)\cdot\tau(W^u_{loc}(p)\cap K)>1}

For sake of concreteness, let’s suppose that {|\det df^n(p)|< 1}. Note that this implies {|\det dg^n(p_g)|< 1} for all {g\in\mathcal{U}}. Denote by {\lambda_g<1<\sigma_g} the eigenvalues of {dg^n(p)}, so that {|\lambda_g\cdot\sigma_g|=|\det dg^n(p_g)|<1}. Now, given {g_0\in\mathcal{U}_+}, we know by Theorem 4that the stable and unstable laminations of {K_{g_0}} meet tangentially at some point in {V}. Since the stable and unstable manifolds of the periodic point {p_{g_0}} are dense in the stable and unstable laminations of {K_{g_0}} (cf. Chapter 2 of Palis-Takens book), one can apply arbitrarily small perturbations to {g_0\in\mathcal{U}_+} so that there is no loss of generality in assuming that {W^s(p_{g_0})} and {W^u(p_{g_0})} meet tangentially at some point in the region {V}. Starting from this quadratic tangency, one gets the following picture for diffeomorphisms {g_{\mu}\in\mathcal{U}_+} close to {g_0}:

Figure 3. Selection of adequate boxes {B_m} for the renormalization of {g_{\mu_m}}.

As it is shown in Chapter 3 of Palis-Takens book (see pages 46–52), one can carefully choose parameters {\mu_m} ({m\in\mathbb{N}}) such that

  • {\mu_m\rightarrow 0} as {m\rightarrow+\infty} and
  • the map {g_{\mu_m}^m|_{B_m}} can be renormalized in such a way that the renormalizations {G_m} of {g_{\mu_m}^m|_{B_m}} {C^2}-converge to the endomorphism {(\widetilde{x},\widetilde{y})\mapsto (\widetilde{y},\widetilde{y}^2)}.

Remark 2 In principle, the parameters {\mu} must vary in some infinite-dimensional manifold in order to {g_{\mu}} parametrize a neighborhood of {g_0}, but for sake of simplicity of the exposition, we will think of this parameter as a real number {\mu\in\mathbb{R}} “measuring” the distance between the line {W^s(p_{g_\mu})\cap V} and the “tip” of the parabola {W^u(p_{g_{\mu}})\cap V} as indicated in Figure 3.

Remark 3 {g_{\mu_m}^m|_{B_m}} can be renormalized” means that one can perform an adequate {\mu_m}dependent change of coordinates {\phi_{\mu_m}} on {g_{\mu}^m|_{B_m}} to get a new dynamics {G_m=\phi_{\mu_m}^{-1}\circ g_{\mu}^m|_{B_m}\circ\phi_{\mu_m}}.

Note that the diffeomorphisms {G_m} 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 {p} is dissipative: in fact, the area-contraction condition {|\det df^n(p)|<1} says that {g_{\mu_m}^m} become strongly area-contracting as {m\rightarrow\infty} and consequently {g_{\mu_m}^m|_{B_m}} converges (after appropriate “scaling”/renormalization) to a curve and {G_m} converges to an endomorphism of this curve as {m\rightarrow\infty}.

Next, we observe that the endomorphism {(\widetilde{x},\widetilde{y})\mapsto (\widetilde{y},\widetilde{y}^2)} has an attracting fixed point at {(\widetilde{x},\widetilde{y})=(0,0)}. Therefore, by {C^2} convergence of {G_m} towards this endomorphism, we conclude that {g_{\mu_m}^m} has an attracting fixed point in {B_m} for all {m} sufficiently large. In other words, {g_{\mu_m}} has a sink (attracting periodic point) in the region {V} for all {\mu_m} sufficiently small.

This last statement can be reformulated as follows. For each {m\in\mathbb{N}}, denote by {R_m=\{g\in\mathcal{U}_+: g \textrm{ has } m \textrm{ sinks}\}}. Note that {R_m} is open for all {m\in\mathbb{N}} (because any sink is persistence under small perturbations of the dynamics). Moreover, since {g_0\in\mathcal{U}_+} was “arbitrary” in the previous argument, we also have that {R_1} is dense in {\mathcal{U}_+}.

At this stage, the idea of S. Newhouse is to iterate this argument to show that the set

\displaystyle R_{\infty}:=\bigcap\limits_{m\in\mathbb{N}} R_m

of diffeomorphisms of {\mathcal{U}_+} with infinitely many sinks is residual in Baire sense(and, in particular, {R_{\infty}} is dense in {\mathcal{U}_+}). Since {R_m} is open in {\mathcal{U}_+} for all {m\in\mathbb{N}} and {R_1} is dense in {\mathcal{U}_+}, it suffices to prove that {R_{m+1}} is dense in {R_m} for all {m\in\mathbb{N}} to conclude that {R_{\infty}} is residual.

In this direction, one starts with {g_0\in R_m} with {m} periodic sinks {\mathcal{O}_1(g_0),\dots,\mathcal{O}_m(g_0)}. By Theorem 4, we know that the stable and unstable laminations of {K_{g_0}} meet tangentially somewhere in {V}. Since {W^s(p_{g_0})}, resp. {W^u(p_{g_0})}, is dense in the stable, resp. unstable, lamination of {K_{g_0}}, we can assume (up to performing an arbitrarily small perturbation on {g_0}) that {W^s(p_g)} and {W^u(p_g)} meet tangentially at some point {q_{g_0}\in V} and {g_0} has {m} periodic sinks. Next, we select {T} a small neighborhood of {q_{g_0}} such that none of the periodic sinks passes through {W}, i.e., {W\cap \mathcal{O}_i(g_0)=\emptyset} for each {i=1,\dots,m}. By repeating the “renormalization” arguments above (with {V} replaced by {T}), one can produce a sequence of diffeomorphisms {(g_{\mu_j})_{j\in\mathbb{N}}} converging to {g_0} as {j\rightarrow\infty} such that {g_{\mu_j}} has a sink {\mathcal{O}(g_{\mu_j})} passing through {T}. Because the sinks {\mathcal{O}_i(g_{\mu_j})} don’t pass through {T} for all {j} sufficiently large, this means that {\mathcal{O}(g_{\mu_j})} is a new sink of {g_{\mu_j}}, that is, we obtain that {g_{\mu_j}\in R_{m+1}} for all {j} sufficiently large. Since {g_{\mu_j}\rightarrow g_0} as {j\rightarrow\infty}, we conclude that {R_{m+1}} is dense in {R_m}.

In resume, we gave a sketch of the proof of the following result:

Theorem 5 (S. Newhouse) Let {k\in\mathbb{N}}, {k\geq 2}, and let {f\in\mathcal{U}_0} be a {C^k}-diffeomorphism such that

  • the periodic point {p} is dissipative, say {|\det df^n(p)|< 1} where {n} is the period of {p}, and
  • {\tau(W^s_{loc}(p)\cap K)\cdot\tau(W^u_{loc}(p)\cap K)>1}

Then, the subset {R_{\infty}\subset\mathcal{U}_+} 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 {\mathcal{U}_+} is the so-called Newhouse phenomena. This theorem is very important because it says that for a topologically big (residual) set {R_{\infty}} of diffeomorphisms the dynamics is so complicated that there are infinitely many attractors. Thus, if we pick at random a point {x} of {U\cup V}, it is very hard to decide (from the computational point of view for instance) the future of the orbit of {x} 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 {g\in\mathcal{U}_+}. At this point, since we know that it is too naive to try to dynamically describe all {g\in\mathcal{U}_+}, we can ask:

What about the local dynamical behavior of most {g\in\mathcal{U}_+} (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 {p} (of saddle-type) of a {C^2}-diffeomorphism {f} having some point {q} of tangency between {W^s(p)} and {W^u(p)}, S. Newhouse can construct (by further analyzing the renormalization argument described above) open sets {\mathcal{U}} arbitrarily close to {f} such that the subset of diffeomorphisms of {\mathcal{U}} with infinitely many sinks is residual in {\mathcal{U}}.

Remark 5 The attentive reader certainly noticed that we insisted that Newhouse phenomena (Theorem 5) concerns {C^2}-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 {C^2}-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 {K^s(g)} and {K^u(g)} intersect for all {g\in\mathcal{U}_+}) and the reader may be curious whether the Newhouse phenomena occurs for {C^1}-diffeomorphisms. As a matter of fact, it is known that the thickness of {C^1}-regular Cantor sets is not continuous, so that the Newhouse gap lemma can’t be applied in the {C^1}-context. Of course, it could be that {C^1}-regular Cantor sets intersect often/stably, thus giving some hope for an analog of Newhouse’s thickness mechanism to survive in the {C^1}-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 {C^2}-setting but still the {C^1}-generic element of the family has finitely many sinks.

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Responses

  1. Hello Carlos,
    a very nice post, I look forward for the next ones.
    Just a question on the proof of Lemma 2; you say that the convex hulls {I} and {I’} have to be linked. I don’t see why {I\subset\I’} or {I’\subset\I} are not possible and in fact I don’t think that it is used in the rest of the proof.
    Cordially.

  2. Hello Quentin,

    You’re absolutely right! In particular, after a little edition, this part of the argument is now correct.

    Best regards,

    Carlos Matheus


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