Some recent results on dynamics of the standard map will be discussed. In particular, we prove that stochastic sea of the standard map has full Hausdorff dimension for sufficiently large topologically generic parameters.

What kind of dynamical phenomena appear after a homoclinic bifurcation of an area preserving diffeomorphism? First we will remind some known results (mostly by P.Duarte) on conservative Newhouse phenomena and properties of the standard map, and then explain how those results can be improved to get a better understanding of the conservative Newhouse phenomena. In particular, we will show that stochastic sea of the standard map has full Hausdorff dimension for sufficiently large topologically generic parameters (a well-known open conjecture claims that it has positive measure).

Given a diffeomorphism of a two-dimensional manifold with a class of smoothness greater than one. Given a horseshoe of this diffeomorphism, R. Mane (1979) showed, based partially on a program introduced by R. Bowen (1973), that the Hausdorff dimension of this horseshoe depends smoothly on the diffeomorphism. We shall give a general discussion of Mane's aforementioned paper, and the techniques used therein.

Jayne's maximum entropy principle is a widely used method for learning probabilistic models of data. Learning the parameters of such models is computationally intractable for most problems of interest in machine learning. As a result one has to resort to severe approximations. However, by "appropriately tweaking" the standard learning rules, one can define a nonlinear dynamical system without fixed points or even periodic orbits.This system is related to a family of weakly chaotic systems known as "piecewise isometries" which have vanishing topological entropy. The symbolic sequences of the very simplest 1 dimensional system areequivalent to Sturmian sequences. The averages over the symbolic sequences of many coupled variables can be shown to capture the relevant correlations present in the data. In this sense, we use this system to learn from data and make new predictions.