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.