This article is a survey of recent results on dimension of repellers for expanding maps and limit sets for iterated function systems. While the case of conformal repellers is well understood the study of nonconformal repellers is in its early stages though a number of interesting phenomena have been discovered, some remarkable results obtained and several interesting examples constructed. We will describe contemporary state of the art in the area with emphasis on some new emerging ideas and open problems.

We demonstrate essential coexistence of hyperbolic and non-hyperbolic behavior in the continuous-time case by constructing a smooth volume preserving flow on a five-dimensional compact smooth manifold that has non-zero Lyapunov exponents almost everywhere on an open and dense subset of positive but not full volume and is ergodic on this subset while having zero Lyapunov exponents on its complement. The latter is a union of three-dimensional invariant submanifolds, and on each of these submanifolds the flow is linear with Diophantine frequency vector.

Dedicated to the memory of Nikolai Chernov.Abstract. We investigate a wide class of two-dimensional hyperbolic systems with singularities, and prove the almost sure invariance principle (ASIP) for the random process generated by sequences of dynamically Hölder observables. The observables could be unbounded, and the process may be non-stationary and need not have linearly growing variances. Our results apply to Anosov diffeomorphisms, Sinai dispersing billiards and their perturbations. The random processes under consideration are related to the fluctuation of Lyapunov exponents, the shrinking target problem, etc.2010 Mathematics Subject Classification. 37D50, 37A25, 60F17.

We construct an example of a Hamiltonian flow f t on a 4-dimensional smooth manifold M which after being restricted to an energy surface M e demonstrates essential coexistence of regular and chaotic dynamics that is there is an open and dense f tinvariant subset U ⊂ M e such that the restriction f t |U has nonzero Lyapunov exponents in all directions (except the direction of the flow) and is a Bernoulli flow while on the boundary ∂U , which has positive volume all Lyapunov exponents of the system are zero.