Birkhoff Spectra of symbolic almost one-to-one extensions

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.

💡 Research Summary

The paper investigates the set of possible visiting frequencies (the Birkhoff spectrum) for points of a dynamical system when observed on a “nice’’ set C, focusing on the case where C is a fat Cantor set (nowhere dense but of positive Lebesgue measure). While classical results guarantee that for sets with null boundary the Birkhoff average exists and equals the Lebesgue measure for every point, the situation for fat sets is largely unknown; in particular, no explicit example of a fat Cantor set with a completely described Birkhoff spectrum has been known.

To address this gap, the author works in the symbolic setting of Cantor minimal systems, which can be represented by Bratteli‑Vershik diagrams. The key technical tool introduced is a “copy‑pasting’’ construction: starting from a Bratteli diagram B, one builds a new diagram ˆB by replicating vertices and edges while preserving the order structure. This yields an almost‑one‑to‑one (a.e. 1‑to‑1) factor map π : (ˆΣ,σ) → (Σ,σ) whose fibers have uniformly bounded cardinality. The construction makes it possible to control the combinatorial structure of the extension and, consequently, the set of invariant measures that give positive weight to a distinguished set D⊂Σ.

The first main result (Theorem A) states that if the extension has fibers of size at most two, then the Birkhoff spectrum S_D of the set D is the full interval