Year: 2026
Type: article
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 ∈ 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 lim n → ∞ 1 / n ⋅ ∑ i = 0 n − 1 1 C ( f i ( x ) ) exists and equals Leb T 1 ( 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.
Source: Advances in Mathematics
Author: Gabriel Fuhrmann
Institution: Durham University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,790
OpenAlex ID: W4417457145
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