Topological Pressure for the Completely Irregular Set of Birkhoff Averages

TL;DR

This paper investigates the topological pressure of irregular sets in dynamical systems, showing that under certain properties, these sets are large, dense, and carry full pressure, extending to multiple observables and various systems.

Contribution

It generalizes the study of Birkhoff averages to multiple observables and establishes the largeness of irregular sets in systems with g-almost product and separation properties.

Findings

Jointly-irregular sets are either empty or have full topological pressure.

Non-uniquely ergodic systems have nonempty, dense, full-pressure completely-irregular sets.

Results apply to various systems like shifts, hyperbolic diffeomorphisms, and flows.

Abstract

In this paper we mainly study the dynamical complexity of Birkhoff ergodic average under the simultaneous observation of any number of continuous functions. These results can be as generalizations of [6,35] etc. to study Birkhorff ergodic average from one (or finite) observable function to any number of observable functions from the dimensional perspective. For any topological dynamical system with $g-$almost product property and uniform separation property, we show that any {\it jointly-irregular set}(i.e., the intersection of a series of $\phi-$irregular sets over several continuous functions) either is empty or carries full topological pressure. In particular, if further the system is not uniquely ergodic, then the {\it completely-irregular set}(i.e., intersection of all possible {\it nonempty $\phi-$irregular} sets) is nonempty(even forms a dense $G_\delta$ set) and carries full topological pressure. Moreover, {\it irregular-mix-regular sets} (i.e., intersection of some $ \phi-$irregular sets and $ \varphi-$regular sets) are discussed. Similarly, the above results are suitable for the case of BS-dimension. As consequences, these results are suitable for any system such as shifts of finite type or uniformly hyperbolic diffeomorphisms, time-1 map of uniformly hyperbolic flows, repellers, $\beta-$shifts etc..