Type: Seminar

Distribution: World

Expiry: 6 Jun 2008

Auth: jonh@asti.maths.usyd.edu.au

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.