A dynamical interpretation of Patterson-Sullivan distributions

TL;DR

This paper explores the relationship between Patterson-Sullivan distributions and dynamical zeta functions in higher-dimensional hyperbolic spaces, linking geometric, dynamical, and quantum ergodic properties.

Contribution

It extends the understanding of Patterson-Sullivan distributions and their connection to zeta functions from surfaces to higher-dimensional hyperbolic spaces.

Findings

Patterson-Sullivan distributions are asymptotically equivalent to Wigner distributions.

The work generalizes previous surface case results to higher dimensions.

Establishes a link between phase space distributions and dynamical zeta functions.

Abstract

Given a compact real hyperbolic space we study the connection between certain phase space distributions, so called Patterson-Sullivan distributions, and dynamical zeta functions. These zeta functions generalize logarithmic derivatives of classical Selberg zeta functions which are defined by closed geodesics which is data from the geodesic ow on phase space. Patterson-Sullivan distributions are asymptotically equivalent to Wigner distributions which play a key role in quantum ergodicity but they are also invariant under the geodesic ow. The surface case was studied before in [AZ07] and thus the emphasis in this work lies on the higher dimensional case.