Borel complexity of sets of points with prescribed Birkhoff averages in Polish dynamical systems with a specification property

TL;DR

This paper investigates the descriptive complexity of sets of points with prescribed Birkhoff averages in Polish dynamical systems, showing many are Borel under compactness and analyzing their hierarchy placement with the specification property.

Contribution

It characterizes the Borel and projective hierarchy complexity of these sets, demonstrating the role of compactness and the specification property in their descriptive complexity.

Findings

Many sets are Borel in compact spaces.

Non-compact spaces can have non-Borel sets at the first projective level.

Sets are located at most at the third level of the hierarchy.

Abstract

We study the descriptive complexity of sets of points defined by placing restrictions on statistical behaviour of their orbits in dynamical systems on Polish spaces. A particular examples of such sets are the set of generic points of a $T$-invariant Borel probability measure, but we also consider much more general sets (for example, $\alpha$-Birkhoff regular sets and the irregular set appearing in multifractal analysis of ergodic averages of a continuous real-valued function). We show that many of these sets are Borel. In fact, all these sets are Borel when we assume that our space is compact. We provide examples of these sets being non-Borel, properly placed at the first level of the projective hierarchy (they are complete analytic or co-analytic). This proves that the compactness assumption is in some cases necessary to obtain Borelness. When these sets are Borel, we use the Borel hierarchy to measure their descriptive complexity. We show that the sets of interest are located at most at the third level of the hierarchy. We also use a modified version of the specification property to show that for many dynamical systems these sets are properly located at the third level. To demonstrate that the specification property is a sufficient, but not necessary condition for maximal descriptive complexity of a set of generic points, we provide an example of a compact minimal system with an invariant measure whose set of generic points is $\Pi^0_3$-complete.