Polynomial growth of volume of balls for zero-entropy geodesic systems

TL;DR

This paper establishes polynomial bounds on the volume growth of balls in universal covers for zero-entropy geodesic systems, introducing new invariants and analyzing specific examples like tori.

Contribution

It introduces strong and weak polynomial entropy invariants and proves polynomial analogues of Manning's inequality for zero-entropy systems.

Findings

Volume growth bounded by strong polynomial entropy

Equality of bounds for flat tori

Explicit asymptotics for revolution tori

Abstract

The aim of this paper is to state and prove polynomial analogues of the classical Manning inequality relating the topological entropy of a geodesic flow with the growth rate of the volume of balls in the universal covering. To this aim we use two numerical conjugacy invariants, the {\em strong polynomial entropy $h_{pol}$} and the {\em weak polynomial entropy $h_{pol}^*$}. Both are infinite when the topological entropy is positive and they satisfy $h_{pol}^*\leq h_{pol}$. We first prove that the growth rate of the volume of balls is bounded above by means of the strong polynomial entropy and we show that for the flat torus this inequality becomes an equality. We then study the explicit example of the torus of revolution for which we can give an exact asymptotic equivalent of the growth rate of volume of balls, which we relate to the weak polynomial entropy.