ABSTRACT:

Given a sequence of random variables $X_0,X_1, \ldots$, Extreme Value Theory is the study of the variable $M_n:=\max\{X_0, X_1,\ldots, X_{n-1}\}$ as $n$ goes to infinity under some suitable scaling. For a dynamical system the random variables are replaced by the values $\phi\circ f^k(x)$ for an observation $\phi$ and a typical point $x$. I will discuss joint work with A.C. and J.M. Freitas on Extreme Value Theory for multimodal interval maps. Our results rely on the fact that there is a close link between Extreme Value Theory and the theory of return time statistics for interval maps with acips. In particular, if a system has exponential return time statistics then Extreme Value laws hold too. This enables us to generalise Collet's work on Gumbel's Law, which he proved for Collet-Eckmann maps, to any multimodal map with an acip. Moreover, we can use Extreme Value Theory to prove that for maps with acips, essentially the only possible return time statistics is exponential.