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.

Dark matter has been a controversial and mysterious topic since 1930s when Zwicky noticed a difference in the amount of mass obtained when computed in different manners. But much of the computations are based on what we knew about the Newtonian N-body problem 70 years ago. In this lecture, more recent results about the dynamics of the Newtonian N-body problem are described; it is shown how these results cast a new "light" on some of the dark matter assertions.

Newhouse stated that unfoldings of homoclinic tangencies of a surface $C^2$-diffeomorphisms yield open sets where the diffeomorphisms with infinitely many sinks/sources are locally generic. There is a version of this result for parametrized families of diffeomorphisms. Palis conjectured that the set of parameters corresponding to diffeomorphisms with infinitely many sinks has measure zero. Gorodetski-Kaloshin gave a partial answer to this conjecture.

Motivated by these results, we study a formulation of this result in the partially hyperbolic setting, where sinks/sources are replaced by homoclinic classes and homoclinic tangencies by heterodimensional cycles. Our result is that it is not possible to generate infinitely many different homoclinic classes using a renormalization-like construction.

This is a join work with J. Rocha (Porto, Portugal).