The Irregular Set for Maps with the Specification Property has Full Topological Pressure

TL;DR

This paper proves that for maps with the specification property, the irregular set of points with non-converging Birkhoff averages either is empty or has full topological pressure, extending understanding of chaotic dynamics.

Contribution

It establishes conditions under which the irregular set has full topological pressure for systems with the specification property, including new examples and applications.

Findings

Irregular set is either empty or has full topological pressure.

Provides conditions characterizing when the irregular set is full.

Shows the irregular set has full topological entropy for suspension flows.

Abstract

Let $(X,d)$ be a compact metric space, $f:X \mapsto X$ be a continuous map with the specification property, and $\varphi: X \mapsto \IR$ a continuous function. We consider the set of points for which the Birkhoff average of $\varphi$ does not exist (which we call the irregular set for $\varphi$) and show that this set is either empty or carries full topological pressure (in the sense of Pesin and Pitskel). We formulate various equivalent natural conditions on $\varphi$ that completely describe when the latter situation holds and give examples of interesting systems to which our results apply but were not previously known. As an application, we show that for a suspension flow over a continuous map with specification, the irregular set carries full topological entropy.