Recurrence rates for observations of flows

TL;DR

This paper investigates recurrence rates in flows, establishing links to local measure dimensions for Anosov flows and providing bounds for observations, with applications to geodesic flows.

Contribution

It introduces new bounds and relationships for recurrence rates in flows and their observations, especially for Anosov and suspension flows.

Findings

Recurrence rates are linked to local measure dimensions in Anosov flows.

Upper bounds for recurrence rates depend on the push-forward measure.

Lower bounds exist for flows with rapidly mixing base dynamics.

Abstract

We study Poincar\'e recurrence for flows and observations of flows. For Anosov flow, we prove that the recurrence rates are linked to the local dimension of the invariant measure. More generally, we give for the recurrence rates for the observation an upper bound depending on the push-forward measure. When the flow is metrically isomorphic to a suspension flow for which the dynamic on the base is rapidly mixing, we prove the existence of a lower bound for the recurrence rates for the observation. We apply these results to the geodesic flow and we compute the recurrence rates for a particular observation of the geodesic flow, i.e. the projection on the manifold.