**Title:** From quantum/wave mechanics to classical mechanics: microlocal analysis of dynamical zeta functions

**Speaker:** Maciej Zworski

**Speaker Info:** Berkeley

**Brief Description:**

**Special Note**:

**Abstract:**

Dynamical zeta functions were introduced by Selberg, Artin--Mazur, Smale and Ruelle. The Ruelle zeta function is defined by replacing primes in the Euler product of the Riemann zeta functions by exponentials of lengths of closed trajectories. Zeta functions, once meromorphically continued, contain information about the distribution of these lengths, the rate of decay to equilibrium and about other properties of the system. Conjectured by Smale in 1967, the meromorphy was proved in 2012 by Giulietti--Liverani--Pollicott for Anosov flows and by Dyatlov--Guillarmou for a class of Axiom A flows in 2014. I will explain a simple microlocal proof of the Anosov case given with Dyatlov in 2013: the key components are a microlocal framework introduced by Faure--Sj\"ostrand 2011, radial propagation results of Melrose 1994, a trace formula of Atiyah--Bott 1967 and Guillemin 1977 and some basic wave front set properties.This reverses the strategy presented in the first lecture: we now use microlocal techniques to analyse purely dynamical questions.

As an application I will present a recent result obtained with Dyatlov: for compact surfaces with Anosov geodesic flows, Ruelle zeta function at 0 has a pole of multiplicity given by the Euler characteristic. In particular, the lengths spectrum (the set of the lenghts of closed geodesics) determines the genus.

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