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∈ X to sufficiently ``nice'' sets C⊆ X under iteration of f. For example, if f is an irrational rotation on the circle, it is well-known that the Birkhoff average limn→∞1/n· ∑ᵢ₌₀ⁿ⁻¹ 1C(fⁱ(x)) exists and equals LebT¹(C) for all x whenever C is measurable with boundary ∂ C of zero Lebesgue measure. If, however, ∂ 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.