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 $Df$invariant 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) (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 $\mathcal{F}^s$:

Here $\mathcal{F}^s$ 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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