Posted by: matheuscmss | November 23, 2008

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

My next topic on my series of posts about some lectures of the School and Workshop of Dynamical held at the ICTP, Trieste (Italy) is a beautiful theorem of Pedro Duarte quoted by Anton Gorodestki during his talk ”On the size of the stochastic layer of the standard map“. I should say that I am not planning to discuss Gorodetski’s theorem here but I will focus only on Duarte’s result. My main reason to do so is the fact that Gorodetski result is strongly based on Duarte’s techniques, so that it would be very hard for me to explain the former theorem without the latter one.

The plan of this post is the following:

  • in next section, we introduce the standard map and briefly discuss the some of its well-known dynamics (due to the KAM theory);
  • after that, we recall the positive entropy conjecture for the standard map;
  • finally, we state Duarte’s theorem showing that the positive entropy conjecture for the standard map is a subtle problem due to the presence of ‘elliptic islands‘.

Closing this introduction, let me point out that I will postpone the proof of Duarte’s theorem to a subsequent discussion.

Standard map: a discrete-time model of the pendulum dynamics

During our undergratude studies in Mathematics (or Physics perhaps), we certainly faced at some point the so-called pendulum equation:

x''(t)=K\sin(2\pi x(t)).

This ODE models an idealized pendulum: the chord is massless and inextensible, the system moves on a 2-dimensional plane due only to the gravitational force and the total energy of the system is conserved (i.e., there is no friction). For more details, see the link above (for the Wikipedia article on the mathematical pendulum).

It is known that the dynamics of the pendulum is very rich. In order to approach it, one can look at the discrete-in-time model provided by the solutions of the following difference equation:

(1) \Delta^2 x_n = (x_{n+1}-x_n)-(x_n-x_{n-1}) = k\sin(2\pi x_n)

obtained by the the substitution of the second order differential operator x''(t) by the second order difference operator \Delta^2 x_n. A simple way to encode the pairs (x_n, x_{n-1}) satisfying the difference equation (1) is provided by the orbits of the so-called standard family (of area-preserving maps acting on the 2-torus \mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2):

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

In fact, it is easy to check that f_k(x_n,x_{n-1})=(x_{n+1},x_n) (exercise).

Remark. Observe that the maps f_{-k} and f_k are conjugated via the translation (x,y)\mapsto (x+\tfrac{1}{2}, y+\tfrac{1}{2}). Hence, we can restrict ourselves to the standard family f_k with parameters k\geq 0.

On the other hand, we should point out that the dynamics of the standard map f_k is a good aproximation of the pendulum dynamics x''(t)=k\sin(2\pi x(t)) only for small values of the parameter k. Indeed, the pendulum’s flow is always integrable (since the total energy, i.e., the kinetic energy plus the potential energy is a conserved quantity for this flow), the same is not exactly true for the standard family: while the standard map f_0 is integrable, that is, \mathbb{T}^2 is foliated by completely invariant (KAM) circles (whose dynamics is conjugated to a rotation), these circles starts to break up as the parameter k increases and the orbit behavior becomes “chaotic”. In fact, this informal description of the dynamics of the standard family for large k is strongly supported by the computer experiments. For instance, one can see below a computer simulation ( borrowed from this Wikipedia article about the standard map) of the 10 orbits of the standard map f_2:

The orbit space of the standard map with parameter k=2. The green region is the ``chaotic'' part of the dynamics (while in the complement of this region the reader can find some invariant KAM circles).

The orbit space of the standard map with parameter k=2. The green region is the “chaotic

The reader can found further beautiful pictures (for other choices of parameters) in this Scholarpedia article. Of course, one can ask (motivated by this conjectural description) the following question:

Question (Positive entropy conjecture). Is the Kolmogorov-Sinai entropy h_\mu(f_k) of the Lebesgue measure \mu positive for large values of k?

While the computer experiments seem to show a lower bound h_{\mu}(f_k)\geq \log(|k|/2), the positive entropy conjecture is not known to be true even for a single value of k (despite the efforts of many mathematicians, e.g., Oliver Knill who tried an operator theory strategy based on the so-called Herman’s subharmonicity method). Anatole Katok describes the positive entropy conjecture as one of the five most resistant problems in dynamics!

The positivity of the (metric) entropy of the standard map f_k for typical large values of k has several interesting consequences:

  • from the Pesin’s entropy formula, we know that the positivity of the entropy of the Lebesgue measure (area) implies the existence of an invariant set A_k of positive area such that every x\in A_k has a positive (maximal) Lyapounov exponent (and, due to the conservation of mass,  a negative minimal Lyapounov exponent also).
  • in particular, from Pesin’s theory, the non-vanishing of the (two) Lyapounov exponents of f_k on A_k (i.e., its non-uniform hyperbolicity) implies that the dynamics of f_k restricted to the set A_k is chaotic (for instance, the system exhibits sensitivity to initial conditions and Katok’s theorem allows us to infer the existence of ‘large’ horseshoes contained inside A_k).

At this point, we are ready to state the main theorem of this post:

Theorem (Duarte). There exists a large parameter k_0>0 and a residual subset R\subset [ k_0,\infty ) such that set E of elliptic periodic points of the standard map f_k is 4/k^{1/3}-dense set (i.e., d(x,E)<4/k^{1/3} for any x\in\mathbb{T}^2) for all k\in R.

It is worthwhile to point out that Duarte’s theorem goes into the opposite direction of the positive entropy conjecture. In fact, the presence of elliptic periodic points is potentially dangerous for the positivity of the entropy because the KAM theory says that a typical elliptic periodic point (i.e., an elliptic point satisfying a certain non-resonance condition) is a full density point of a Cantor set laminated by invariant curves whose dynamics are conjugated to rotations on circles. In particular, the entropy of the initial diffeomorphism restricted to this Cantor set of invariant curves surrounding the elliptic point is zero. In other words, the presence of these ‘elliptic islands’ (i.e., positive measure invariant regions where the diffeomorphism behaves like rotations) are a serious obstruction to the positivity of the entropy.

Remark. Observe that Duarte’s theorem doesn’t contradicts the positive entropy conjecture. In fact, this follows from the ‘usual reason’: residual sets (in Baire sense) may have zero Lebesgue measure. However, this result certainly shows how subtle the positive entropy conjecture is.

Ending this post, let me outline the strategy of the proof of Duarte’s theorem. Roughly speaking, the idea is to get the result from a conservative version of Newhouse phenomena. More precisely, one begins with the following result:

Theorem 1. For each parameter k, there exists a basic set \Lambda_k of f_k (i.e., a compact f_k-invariant transitive hyperbolic set) with the following properties:

  • the family \{\Lambda_k\}_{0\leq k<\infty} is dynamically increasing: given k_0, for any sufficiently small \epsilon>0, the set \Lambda_{k_0+\epsilon} contains the dynamical continuation of \Lambda_{k_0} (that is, the unique compact f_{k_0+\epsilon}-invariant hyperbolic set close to \Lambda_{k_0} in the Hausdorff topology);
  • the thickness of \Lambda_k grows to infinity as k\to\infty: for all sufficiently large parameter k, the local stable and unstable thickness are

\tau^s_{loc}(\Lambda_k), \tau^u_{loc}(\Lambda_k)\geq k^{1/3}/9;

HD(\Lambda_k)\geq 2\log2/\log(2+9k^{-1/3});

  • \Lambda_k is topologically conjugated to a (full) Bernoulli shift in 2n_k symbols where

2n_k/4k\to 1 as k\to\infty;

  • for all large k, the set \Lambda_k is 4/k^{1/3}-dense in the torus \mathbb{T}^2.

Starting with this theorem, Duarte shows that the basic set \Lambda_k generically unfolds a quadratic homoclinic tangency for a dense set of (large) parameters:

Theorem 2. There exists k_0>0 such that, for any k\geq k_0 and any periodic point p\in\Lambda_k, the set \mathcal{T} of parameters \ell\in\left[k,\infty\right) such that the invariant manifolds W^s(p(\ell)) and W^u(p(\ell)) of the continuation p(\ell) of p generically unfold a quadratic homoclinic tangency is dense in \left[ k,\infty \right).

In resume, the theorems 1 and 2 put us in the framework of the (conservative version of) Newhouse phenomena: for a large parameter in the dense set \mathcal{T}, we have a thick basic set unfolding (generically) a quadratic tangency; this permits to use some of the ideas employed by Newhouse (as discussed in a previous post): of course, it is impossible to obtain the same conclusion of Newhouse (namely, the coexistence of infinetely many sinks/sources for a generic nearby system) because we are dealing exclusively with area-preserving diffeomorphisms; but, the techniques of Newhouse gives us the coexistence of many elliptic periodic points. Indeed, this is the content of the following result of Duarte:

Theorem 3. There are k_0>0 large and a residual subset R\subset \left[k_0,\infty\right) such that, for every k\in R, the closure of the elliptic periodic points of f_k contains the 4/k^{1/3}-dense basic set \Lambda_k.

Of course, Duarte’s theorem is an immeadiate consequence of theorem 3. In the next post, we will concentrate on the proofs of theorems 1, 2 and 3. Ja ne!


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