Joint Colloquium: Field -- Dynamical zeta functions and mixing

The dynamical zeta function is an analogue of the Riemann zeta function.
However, rather than being the Euler product over the prime numbers, the
dynamical zeta function is a product over the prime periods of a flow. Just
as happens in number theory, analytic and meromorphic properties of the
dynamical zeta function encapsulate statistical properties of the flow such
as the distribution of periodic orbits (prime number theorem) and rates of
mixing. In this introductory talk we will describe some of the characteristic
properties of dynamical zeta functions. We will also discuss the issue of
exponential error estimates (which correspond to the Riemann hypothesis in
number theory) as well as recent work on rates of mixing for hyperbolic flows
including, we hope, new examples of smooth hyperbolic flows that stably mix
exponentially fast.