Year: 2026
Type: preprint
Abstract: In this paper, we study the structure of Birkhoff spectra for hyperbolic dynamical systems. Given a Hölder observable \(f\) on a basic set \(Λ\), we obtain the following results: First, we characterize when the Birkhoff spectrum of \(f\) is dense in the positive (or negative) real line. Second, we prove that a bounded Birkhoff spectrum forces \(f\) to be cohomologous to zero, which constitutes an extension of Livšic's theorem. Moreover, we show that if the spectrum exhibits an ``arithmetically sparse'' structure, then \(f\) is cohomologous to a constant. \\ \indent We then extend these results to continuous time. For Anosov flows -- including geodesic flows on Anosov manifolds -- we establish analogous density results for Birkhoff integrals over closed orbits. In particular, we generalize a theorem of Dairbekov--Sharafutdinov \cite{Dairbekov} by proving that a bounded (resp.~arithmetically sparse) spectrum forces a smooth function to vanish (resp.~be constant).
Source: arXiv (Cornell University)
Author: Romaña, Sergio
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7125103754
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