The gluing orbit property, uniform hyperbolicity and large deviations principles for semiflows

TL;DR

This paper introduces a new, weaker gluing orbit property for maps and flows, proves its relation to hyperbolicity, and establishes large deviations principles for semiflows with this property.

Contribution

It defines a weaker gluing orbit property, links it to uniform hyperbolicity, and derives large deviations principles for semiflows satisfying this property.

Findings

Flows with the robust gluing orbit property are uniformly hyperbolic.

Large deviations principles are established for semiflows with the gluing orbit property.

Criteria are provided for suspension flows to satisfy the gluing orbit property.

Abstract

In this article we introduce a gluing orbit property, weaker than specification, for both maps and flows. We prove that flows with the $C^1$-robust gluing orbit property are uniformly hyperbolic and that every uniformly hyperbolic flow satisfies the gluing orbit property. We also prove a level-1 large deviations principle and a level-2 large deviations lower bound for for semiflows with the gluing orbit property. As a consequence we establish a level-1 large deviations principle for hyperbolic flows and every continuous observable, and also a level-2 large deviations lower bound. Finally, since many non-uniformly hyperbolic flows can be modeled as suspension flows we also provide criteria for such flows to satisfy uniform and non-uniform versions of the gluing orbit property.