Birkhoff Spectra of symbolic almost one-to-one extensions

TL;DR

This paper investigates the Birkhoff spectra of fat Cantor sets within minimal dynamical systems, revealing that such spectra can be either full intervals or non-intervals, and applies these findings to irrational rotations.

Contribution

It introduces a method to analyze Birkhoff spectra of fat Cantor sets in minimal systems and demonstrates the existence of sets with diverse spectral properties.

Findings

Existence of fat Cantor sets with full Birkhoff spectra as intervals.

Existence of fat Cantor sets with non-interval Birkhoff spectra.

Application to irrational rotations showing diverse spectral behaviors.

Abstract

Given a continuous self-map $f$ on some compact metrisable space $X$, it is natural to ask for the visiting frequencies of points $x\in X$ to sufficiently ``nice'' sets $C\subseteq X$ under iteration of $f$. For example, if $f$ is an irrational rotation on the circle, it is well-known that the Birkhoff average $\lim_{n\to\infty}1/n\cdot \sum_{i=0}^{n-1}\mathbf 1_C(f^i(x))$ exists and equals $\textrm{Leb}_{\mathbb T^1}(C)$ for all $x$ whenever $C$ is measurable with boundary $\partial C$ of zero Lebesgue measure. If, however, $\partial C$ is fat (of positive measure), the respective averages can generally only be evaluated almost everywhere or on residual sets. In fact, there does not appear to be a single example of a fat Cantor set $C$ whose Birkhoff spectrum -- the full set of visiting frequencies -- is known. In this article, we develop an approach to analyse the Birkhoff spectra of a natural class of dynamically defined fat nowhere dense compact subsets of Cantor minimal systems. We show that every Cantor minimal system admits such sets whose Birkhoff spectrum is a full non-degenerate interval -- and also such sets for which the spectrum is not an interval. As an application, we obtain that every irrational rotation admits fat Cantor sets $C$ and $C'$ whose Birkhoff spectra are, respectively, an interval and not an interval.