Escape rates for special flows and their higher order asymptotics

TL;DR

This paper investigates escape rates for special flows, establishing their dependence on the ceiling function, and derives higher order asymptotics under regularity conditions, with implications for ergodic theory.

Contribution

It provides a comprehensive analysis of escape rates for special flows, including monotonicity, scaling, invariance, and higher order asymptotics, extending previous results in ergodic theory.

Findings

Escape rate depends monotonically on the ceiling function.

Local escape rate equals the base escape rate divided by the integral of the ceiling function.

Higher order asymptotics are established under regularity conditions.

Abstract

In this paper escape rates and local escape rates for special flows are sudied. In a general context the first result is that the escape rate depends monotonically on the ceiling function and fulfills certain scaling, invariance, and continuity properties. For the metric setting local escape rates are considered. If the base transformation is ergodic and exhibits an exponential convergence in probability of ergodic sums, then the local escape rate with respect to the flow is just the local escape rate with respect to the base transformation, divided by the integral of the ceiling function. Also a reformulation with respect to induced pressure is presented. Finally, under additional regularity conditions higher order asymptotics for the local escape rate are established.