# Seminar - Center for Applied Mathematics

Dynamical Zeta Functions and Prime Orbit Theorems in Complex Dynamics

Zhiqiang Li , Stony Brook University

Refreshments at 3:45.

Analogues of the Riemann zeta function were first introduced into geometry by A. Selberg and into dynamics by M. Artin, B. Mazur, and S. Smale. Analytic studies of such dynamical zeta functions yield quantitative information on the distribution of closed geodesics and periodic orbits.

We obtain the first Prime Orbit Theorem in complex dynamics outside of hyperbolic maps, for a class of maps called expanding Thurston maps f . More precisely, we show that the number of primitive periodic orbits of f , ordered by a weight on each point induced by a non-constant real-valued Hölder continuous function on S2

satisfying some additional regularity conditions, is asymptotically the same as the logarithmic integral, with an exponentially small error term. Such a result follows from our quantitative study of the holomorphic extension properties of the associated dynamical zeta functions and dynamical Dirichlet series.

In particular, the above result applies to postcritically-finite rational maps whose Julia set is the whole Riemann sphere. Moreover, we prove that the regularity conditions needed here are generic; and for a Lattès map f , a continuously differentiable function satisfies such a condition if and only if it is not cohomologous to a constant. This is a joint work with T. Zheng.