Posted by: matheuscmss | February 16, 2009

Proof of Duarte’s theorem on the abundance of elliptic islands for the standard map

Today we’ll close our current discussion of the standard map with the proof of Duarte’s theorem. As we mentioned earlier, the basic strategy consists into three steps: the construction of a dynamically increasing family of hyperbolic basic sets of saddle-type (“horseshoes”), the existence of a dense set of parameters where  a quadratic tangency is generically unfolded and the construction of elliptic islands from the bifurcation of quadratic tangencies via a conservative version of Newhouse phenomena. More precisely, we are going to show the statements of theorems 1, 2 and 3 of the previous post.

An “increasing” family of hyperbolic basic sets

In order to find hyperbolic sets, we’ll use the well-known invariant cone-field criterion:

Theorem (invariant cone-field criterion). Let f:M\to M be a C^1-diffeomorphism. Consider \Lambda a compact f-invariant set. Assume that there are \mu>1>\lambda, a decomposition T_{\Lambda}M=E\oplus F (not necessarily Df-invariant) and two families C^u\supset F, C^s\supset E of (closed) cones such that

  • Df(x)(C^u(x))\subset int(C^u(x)) and \|Df(x)v^u\|\geq\mu\|v^u\| for all x\in\Lambda and v^u\in C^u(x);
  • Df^{-1}(f(x))(C^s(f(x)))\subset int(C^s(x)) and \|Df^{-1}(f(x))v^s\|\geq\lambda^{-1}\|v^s\| for all x\in\Lambda and v^s\in C^s(f(x)).

Then, \Lambda is a hyperbolic set.

Proof. See corollary 6.4.8 of Katok-Hasselblat book. A sketch of proof goes as follows. Fix a point x\in\Lambda and consider the following two sequences of cones on the tangent space T_xM: Df^n(x)\cdot C^u(f^{-n}(x)) and Df^{-n}(x)\cdot C^s(f^n(x)). Our assumptions implies that these two sequences are nested sequences of closed cones, so that the intersections

E^u=\bigcap\limits_{n\geq 0}Df^n(x)\cdot C^u(f^{-n}(x))

and

E^s=\bigcap\limits_{n\geq 0}Df^{-n}(x)\cdot C^s(f^n(x))

are closed Dfinvariant closed cones. In fact, one can work a bit more (with the facts Df(x)(C^u(x))\subset int(C^u(x)) and Df^{-1}(f(x))(C^s(f(x)))\subset int(C^s(x))) to see that E^u and E^s are vector subspaces with dimensions \dim F and \dim E resp.. Finally, once we know that E^u and E^s are Df-invariant subspaces, our assumptions of expansion (resp. contraction) of vectors belonging to the cone C^u\supset E^u (resp. C^s\supset E^s), we see that E^u is the unstable subspace and E^s is the stable subspace. This completes the sketch. \square

Proposition 1. Consider the invertible area-preserving map

f(x,y)=(-y+\phi(x),x)

of the 2-torus T^2 and let \Lambda be a f-invariant compact set. Assume that there exists \lambda>2 such that |\phi'(x)|\geq\lambda for all (x,y)\in\Lambda. Then, \Lambda is a hyperbolic set (of saddle type).

Proof. Note that

Df = \left(\begin{array}{cc}\phi'(x) & -1 \\ 1 & 0\end{array}\right)\in SL(2,\mathbb{R}).

In particular, the trace of Df verifies |tr Df|\geq \lambda>2 so that the matrices Df are uniformly hyperbolic. In fact, this follows from the fact that the constant cone-field

C^u(p)\equiv C_{a}^u:=\{(u,v)\in\mathbb{R}^2: |v|\leq a |u|\}

is an unstable cone-field whenever 1/(\lambda-1)<a<1 (note that such a choice is possible since \lambda>2). Indeed, if we write Df(u,v):=(u',v'), we see that

|v'|= |u|\leq (\lambda-a)^{-1}|\phi'(x)u-v|=(\lambda-a)^{-1}|u'|

so that Df(C_a^u)\subset C_{(\lambda-a)^{-1}}^u = C_{\theta a}^u where \theta = \left(a\cdot(\lambda-a)\right)^{-1}<1 by the choice of the parameter a, i.e., C_a^u is Df-invariant. Furthermore, denoting by \|(u,v)\|=\max\{|u|,|v|\}, we get, for any (u,v)\in C^u_a,

\|Df(u,v)\| = |u'|\geq (\lambda-a)|u| = (\lambda-a)\|(u,v)\|

with (\lambda-a)>1, i.e., Df (uniformly) expands any vector inside C^u_a. On the other hand, it is not hard to see that Df\in SL(2,\mathbb{R}) implies that the same argument can be applied to Df^{-1} in order to get a stable cone-field. Using the invariant cone-field criterion, the proof is complete. \square

An immediate consequence of this proposition is the following result:

Corollary 1. For the standard family

f_k(x,y)=(-y+2x+k\sin(2\pi x),x):=(-y+\phi_k(x),x),

given any \lambda>2, the maximal invariant set

\Lambda_k=\bigcap\limits_{n\in\mathbb{Z}}f_k^{-n}(\{(x,y)\in\mathbb{T}^2: |\phi_k'(x)|\geq\lambda \})

is hyperbolic.

Remark. It is worth to note that this result gives a clue about the location of the critical region of non-hyperbolicity: for a given \lambda>2, the set of points \{(x,y): |\phi_k'(x)|\leq\lambda\}  converges to the union of the two circles \{x=\pm 1/4\} when k\to\infty.

Of course, this corollary says that the hyperbolic sets \Lambda_k are a family of dynamically increasing basic sets. In fact, it turns out that this can be checked by hand (see section 4.2 of Duarte’s paper), but we’ll skip this fact for sake of brevity of the exposition.

Global dynamical foliations and their tangency lines

After the description of the (“big”) hyperbolic sets \Lambda_k of the standard map f_k, we proceed to the study of the tangencies between their invariant foliations. In order to do so, we need to extend these foliations to some uniform neighborhood of \Lambda_k (since we want to perform an analysis for several large parameters) while keeping good estimates of distortion of the holonomy maps. At this point, our first technical problem arises: from the general theory of uniformly hyperbolic sets (see the book of Palis-Takens), we know that \Lambda_k admits some neighborhood U_k so that the stable and unstable foliations of \Lambda_k can be extended to U_k (while keeping good estimates), but a priori the region U_k where the good estimates are ensured can deteriorate when k\to\infty. To overcome this problem, Duarte takes the following point of view. Near the critical region \{x=\pm 1/4\}, he replaces \phi_k(x)=2x+k\sin(2\pi x) by a function \psi_k(x) having two poles at x=\pm 1/4 and he tries to compare the dynamics of f_k(x,y)=(-y+\phi_k(x),x) with the dynamics of the singular diffeomorphism g_k(x,y) = (-y+\psi_k(x),x).

More precisely, \psi_k(x)=\phi_k(x)+\rho_k(x) where \rho_k(x) = 0 outside a 2/k^{1/3}-neighborhood of x=\pm 1/4 and \rho_k(\pm 1/4)=\infty. Then, after the somewhat tedious work of redoing the theory of invariant manifolds (following the exposition of Hirsch-Pugh-Shub), he checks that \psi_k has global stable and unstable foliations \mathcal{F}^s, \mathcal{F}^u on \mathbb{T}^2 verifying uniform distortion estimates (i.e., their holonomy maps have uniformly bounded C^2-norm). Here, the uniform control of distortion comes from the choice of \rho_k: indeed, assuming that \psi_k coincides with \phi_k outside a 1/k^{\epsilon}-neighborhood of x=\pm 1/4, it is not hard to see that the Schwartzian derivate of \psi_k is bounded from below (in the critical region |x\mp 1/4|\leq k^{-\epsilon}) by

2\left|\frac{\psi_k''(x)^2}{\psi_k'(x)^3}\right|+\left|\frac{\psi_k'''(x)}{\psi'(x)^2}\right|\geq \frac{\pi}{k|\cos(2\pi x)|^3} - \frac{\pi}{k\cos(2\pi x)}\geq k^{3\epsilon-1}.

In particular, since we want to take \epsilon>0 the largest possible so that \psi_k coincides with \phi_k in |x\mp 1/4|\geq k^{-\epsilon} and \psi_k with bounded Schwartzian derivative (because it is well-known that bounded Schwartzian derivative implies bounded distortion), it is natural to take \psi_k=\phi_k outside |x\mp 1/4|\geq 2k^{-1/3} (i.e., \epsilon=1/3).

Of course, once we performed this work (which takes 21 pages of Duarte’s paper), we have to compare the dynamics of f_k and g_k. However, this is not hard: the maximal invariant set \Lambda_k is the same for both f_k and g_k, and, using the strong hyperbolic features of the singular diffeomorphism g_k, it is possible to show that the dynamics of f_k and g_k on \Lambda_k are conjugated to a 2n_k full (Bernoulli) shift (where 2k(1-32\pi^2/k^{2/3})\leq n_k\leq 2k); furthermore, B_{4/k^{1/3}}(\Lambda_k)=\mathbb{T}^2,  its stable and unstable thickness \tau^s(\Lambda_k),\tau^u(\Lambda_k)\geq k^{1/3}/9 (where \tau^s(\Lambda_k), resp. \tau^u(\Lambda_k), is the thickness of the Cantor set obtained by projection of \Lambda_k along the stable, resp. unstable, foliation on an arbitrarily fixed transversal section) and, as a consequence, its Hausdorff dimension HD(\Lambda_k) satisfies HD(\Lambda_k)\geq 2 \log 2/\log(2+9/k^{1/3}).

Next, we analyse the relative positions of the (g_k-invariant) foliations \mathcal{F}^s and \mathcal{F}^u. Applying f_k to \mathcal{F}^u, we obtain a new foliation \mathcal{G}^u := (f_k)_*(\mathcal{F}^u) (recall that \mathcal{F}^u is g_k-invariant but it is not f_k-invariant). It is not hard to see that the set of tangencies between \mathcal{G}^u and \mathcal{F}^s are two circles close to \{x=\pm 1/4\}. Moreover, the projection of \Lambda_k along \mathcal{G}^u and \mathbb{F}^s into these two circles gives rise to two Cantor sets K^u and K^s satisfying \tau(K^u)\geq k^{1/3}/10 and \tau(K^s)\geq k^{1/3}/9 (here we are using the previous thickness estimate and the fact that the application of f_k to the foliation \mathcal{F}^u doesn’t change very much the thickness). The picture below (borrowed from Duarte’s paper) summarizes our discussion about the relative position of \mathcal{G}^u and \mathcal{F}^s:

duarte-fig1Here \mathcal{F}^s is the almost vertical foliation and \mathcal{G}^u is the foliation folding along the two dotted circles. Now, using the fact that \tau(K^u)\tau(K^s)\geq k^{2/3}/90\gg 1 (for any large k), we can apply Newhouse’s gap lemma to obtain that K^u\cap K^s\neq\emptyset. In other words, we get that \mathcal{G}^u and \mathcal{F}^s exhibits persistent tangencies.

Remark. In proposition 16 of Duarte’s paper, a version of Newhouse’s gap lemma in the circle is wrongly stated: indeed, Duarte claims that the fact that the two Cantor sets are contained in the circle automatically implies that the two Cantor sets are linked. However, this is not correct (as the example of two thick Cantor sets supported by two disjoint compact intervals shows), although this is not a serious problem for this argument: from a careful checking of the geometry of \Lambda_k (via the features of the singular diffeomorphism g_k), it is not hard to see that K^u and K^s are linked (this follows from Duarte’s argument in section 4.2 of his paper).

Finally, closing this section, we claim that these persistent tangencies are quadratic and unfold generically with the parameter k (as the picture above indicates). While a complete proof of this result takes 7 pages of technical calculations, we’ll provide a convincingly enough (I hope! :)) heuristic argument. We know that the \psi_k-invariant foliations \mathcal{F}^s and \mathcal{F}^u are almost vertical and horizontal (resp.). In particular, it is reasonable to expect that the circles of tangencies between \mathcal{F}^s and \mathcal{G}^u = (f_k)_*(\mathcal{F}^u) are close to the circles of tangencies between the horizontal foliation and the image of the vertical foliation under f_k. On the other hand, since f_k(x,y) = T_k\circ R where R(x,y)=(-y,x) is the \pi/2 counterclockwise rotation and T_k(x,y)=(x+\phi_k(y),y) is a shear (of variable intensity) along the horizontal foliation, we can compute the image of the vertical foliation under f_k as follows: the image of the horizontal foliation by R is the vertical foliation and the image of the vertical foliation by T_k is a foliation by the family of curves which are parallel to the graph \{(\phi_k(y),y):y\in\mathbb{T}\}. In particular, the circle of tangencies between these two foliations are exactly the critical circles \{(x,\nu_{\pm}): x\in\mathbb{T}, \phi'_k(\nu_{\pm})=0\}. At such points, the difference between the curvatures is measured by \phi_k''(\nu_{\pm})=4\pi^2 k, so that the tangencies between the horizontal foliation and the f_k-image of the vertical foliation are quadratic (i.e., locally you are seeing the intersections between straight lines and parabolas) and, a fortiori, the same holds for the tangencies between \mathcal{G}^u and \mathcal{F}^s. Also, when the parameter k increases, the g_k-invariant foliations \mathcal{F}^u and \mathcal{F}^s doesn’t change very much (they are almost constant), while the x-coordinates of the tangency points between \mathcal{G}^u=(f_k)_*(\mathcal{F}^u) and \mathcal{F}^s (which are close to (x+\phi_k(\nu_{\pm}),\nu_{\pm})) move with velocity (close to) 1 (indeed, \nu_{\pm}=\pm 1/4 implies \frac{d}{dk}(x+\phi_k(\nu_{\pm}))= \sin(2\pi \nu_{\pm})=1). Hence, these tangencies are unfolded generically.

At this point, the reader noticed that this discussion gives the theorems 1 and 2 of the previous post. Now, we proceed to discuss the conservative version of the Newhouse phenomena.

-Conservative version of Newhouse phenomena: proof of theorem 3-

Before entering into the proof of the abundance of elliptic islands close to a generically unfolded quadratic tangency, let me review a little bit some facts around the proof of the “classical” Newhouse phenomena.

Given f_\mu a 1-parameter family of surface diffeomorphisms generically unfolding a quadratic homoclinic tangency q associated to a hyperbolic periodic point p of saddle-type (at the parameter \mu=0 say), Newhouse manage to define a renormalization scheme near q as follows: for every large n\in\mathbb{N}, one can select small boxes near q which are mapped by f_\mu^n near itself  with the shape of a parabola so that their relative positions resembles a horseshoe; next, we compose this dynamics with appropriate rescalings h_n of these boxes in order to put these very small boxes into a fixed scale (e.g., a unit square) so that we obtain the families of dynamical systems given by h_n\circ f_\mu^n\circ h_n^{-1} (these are called the successive renormalizations of the dynamics near the homoclinic tangency). The usefulness of idea is more or less clear: assuming that there exists some limiting object h_n\circ f_\mu\circ h_n^{-1}\to t_{\mu}, any stable dynamical property of t_\mu will be shared by h_n\circ f_\mu\circ h_n^{-1} and a fortiori f_\mu.  It turns out that Newhouse showed that this renormalization scheme converges (i.e., the limiting object exists) when the periodic point p is dissipative (i.e., \det|Df_\mu(p)|<1).  Moreover, the limit t_\mu in this case is the quadratic family

t_\mu(x,y)=(\mu-x^2,x).

Using this information, the existence of sinks near the homoclinic tangency follows directly. A detailed exposition of Newhouse’s argument can be found in the excelent book of Palis and Takens.

After this quick review of Newhouse arguments, let us consider again the situation of the standard map: in the conservative setting, given a family of area-preserving diffeomorphisms generically unfolding a quadratic homoclinic tangency associated to a hyperbolic periodic point, one can repeat the renormalization scheme of Newhouse to get as a limit object the conservative Hénon family

t_\mu(x,y)=(-y+\mu-x^2,x).

Next, it is possible (exercise) to show the presence of elliptic fixed points for this family when -1<\mu<3 (with the eigenvalue of this point running from 1 to -1). Because generic (i.e., non-resonant) elliptic periodic point is stable by conservative perturbations (this follows from the so-called KAM theory; see e.g. this monograph of J. Moser), we conclude the existence of (generic) elliptic periodic points nearby the homoclinic tangency. Combining this result with the theorems 1 and 2 proved in the previous sections, we see that, similarly to the proof of Newhouse theorem, the proof of theorem 3 is complete!


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