A Large deviation and an escape rate result for special semi-flows

TL;DR

This paper establishes a large deviation principle and an escape rate estimate for smooth flows built over subshifts of finite type, providing explicit formulas and bounds for these probabilistic behaviors.

Contribution

It introduces explicit formulas for large deviations and escape rates in semi-flows over subshifts, advancing understanding of their statistical properties.

Findings

Explicit large deviation bounds for smooth flows.

Lower bounds for escape rates through small holes.

Quantitative estimates for probabilistic behavior of semi-flows.

Abstract

In this paper we consider a smooth flow $(\Lambda,\Phi^t)$ builded from suspending over a (non-invertible topologically mixing) subshift of finite type, and we equip it with an equilibrium measure $\nu$ on $\Lambda.$ The two main theorems are a large deviation and an escape rate result. The first theorem gives an explicit formula for $X>0$ and $Y$ such that $$\nu\left\{x\in\Lambda: \left|\int F\circ \Phi^s (x) ds-\int F d\mu\right|>\epsilon\right\}\leq \exp(-Xt+\log t+Y)$$ for $t\gg>1\gg\epsilon>0,$ where $F:\Lambda\to\mathbb{R}$ is smooth. The second theorem gives an explicit lower bound for the asymptotic behaviour of the escape rate of $\nu$ through a small hole.