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:
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:
obtained by the the substitution of the second order differential operator by the second order difference operator . A simple way to encode the pairs 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 ):
In fact, it is easy to check that (exercise).
Remark. Observe that the maps and are conjugated via the translation . Hence, we can restrict ourselves to the standard family with parameters .
On the other hand, we should point out that the dynamics of the standard map is a good aproximation of the pendulum dynamics only for small values of the parameter . 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 is integrable, that is, is foliated by completely invariant (KAM) circles (whose dynamics is conjugated to a rotation), these circles starts to break up as the parameter increases and the orbit behavior becomes “chaotic”. In fact, this informal description of the dynamics of the standard family for large 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 :
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 of the Lebesgue measure positive for large values of ?
While the computer experiments seem to show a lower bound , the positive entropy conjecture is not known to be true even for a single value of (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 for typical large values of 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 of positive area such that every 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 on (i.e., its non-uniform hyperbolicity) implies that the dynamics of restricted to the set 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 ).
At this point, we are ready to state the main theorem of this post:
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 , there exists a basic set of (i.e., a compact -invariant transitive hyperbolic set) with the following properties:
- the family is dynamically increasing: given , for any sufficiently small , the set contains the dynamical continuation of (that is, the unique compact -invariant hyperbolic set close to in the Hausdorff topology);
- the thickness of grows to infinity as : for all sufficiently large parameter , the local stable and unstable thickness are
- the Hausdorff dimension of goes to as : for all sufficiently large parameter , it holds
- is topologically conjugated to a (full) Bernoulli shift in symbols where
- for all large , the set is -dense in the torus .
Starting with this theorem, Duarte shows that the basic set generically unfolds a quadratic homoclinic tangency for a dense set of (large) parameters:
Theorem 2. There exists such that, for any and any periodic point , the set of parameters such that the invariant manifolds and of the continuation of generically unfold a quadratic homoclinic tangency is dense in .
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 , 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 large and a residual subset such that, for every , the closure of the elliptic periodic points of contains the -dense basic set .
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!