Limit theorems in dynamical systems

Abstract

The idea of unpredictability was introduced into dynamical systems by Poincaré at the end of 19th century. In form of differential equations, he showed that a completely deterministic system may exhibit features of randomness due to lackof information. Since 1980s there is growing interest to understand probabilistic phenomenon in dynamical systems, thus following Poincaré's ideas. On the other hand, limit theorems themselves have central roles in probability theory, for "large-scale random phenomena in their collective action create strict, nonrandomregularity". My research studies probabilistic limit theorems in dynamical systems, for example distribution of return times and fluctuations of ergodic sums(central limit theorems, local limit theorems, invariance principle etc.) for Gibbs-Markov maps and for (random) dynamical systems derived from them. The type of dynamical system where such results hold is not restricted to Gibbs-Markov dynamics, and many systems can be studied from there.