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.

Normally hyperbolic smooth trapped sets are structurally
stable and occur in many interesting situations: for instance
for Kerr black hole metrics. We show that the corresponding
semiclassical resonances (e.g. quasinormal modes for Kerr
black holes) are separated from the real axis which has
consequences for decay of waves and other phenomena.

The proof is a simple example of techniques
used in the semiclassical study of quantum resonances
and I hope to present it in a self-contained way.

A dynamical system is chaotic if its behavior is sensitive to a change in the initial data. This is usually associated with instability of trajectories. The hyperbolic theory of dynamical systems provides a mathematical foundation for the paradigm that is widely known as "deterministic chaos" -- the appearance of irregular chaotic motions in purely deterministic dynamical systems. This phenomenon is considered as one of the most fundamental discoveries in the theory of dynamical systems in the second part of the last century. The hyperbolic behavior can be interpreted in various ways and the weakest one is associated with dynamical systems with nonzero Lyapunov exponents.

I will describe main types of hyperbolicity and the still-open problem of whether dynamical systems with nonzero Lyapunov exponents are "typical" in a sense. I will outline some recent results in this direction and relations between this problem and two other important problems in dynamics: whether systems with nonzero Lyapunov exponents exist on any phase space and whether nonzero exponents can coexist with zero exponents in a robust way.