# Meeting Details

Title: Metastability in ergodic theory Center for Dynamics and Geometry Seminars Jeremy Avigad, Carnegie Mellon In dynamical systems and ergodic theory, one tries to characterize the behavior of systems that evolve over time. These fields, however, often use methods that are difficult to interpret in computable or quantitative terms. For example, let $T$ be a measure-preserving transformation of a space $(X, \mathcal B, \mu)$, let $f$ be a measurable function from $X$ to $\mathbb{R}$, and for every $x \in X$ and $n \in \mathbb{N}$ let $(A_n f)x = (f x + f(T x) + \ldots + f(T^{n-1} x)) / n$. The pointwise ergodic theorem says that this sequence of averages converges for almost every $x$, and the mean ergodic theorem says that the sequence $(A_n f)$ converges in the $L^2$ norm. But, in general, one cannot compute a rate of convergence from the initial data. Describing joint work with Philipp Gerhardy and Henry Townser, I will explain how proof-theoretic methods provide classically equivalent formulations of the ergodic theorems which are computably valid, and yield additional information. Roughly, the additional information amounts to quantitative bounds on how long one has to wait to find long pockets of stability in the sequence of ergodic averages. These bounds are, moreover, independent of the measure-preserving system. The fact that such bounds can often be obtained is an instance of a phenomena that Terence Tao has called metastability.'' As a second example, I will discuss the Furstenberg-Zimmer structure theorem, which shows that any measure preserving system can be decomposed into to a transfinite sequence of compact extensions terminating with the maximal distal factor. I will describe a metastable version of this theorem, obtained jointly with Towsner, which shows that factors approximating the behavior of the maximal distal factor can occur much earlier in the sequence.