On the Birkhoff Spectrum for Hyperbolic Dynamics

TL;DR

This paper investigates the structure of Birkhoff spectra in hyperbolic dynamical systems, revealing conditions for density, cohomology to zero or constants, and extending results to continuous-time flows including geodesic flows.

Contribution

It extends Livšic's theorem by linking bounded spectra to cohomology and generalizes spectral density and sparsity results to Anosov flows and geodesic flows.

Findings

Birkhoff spectrum density characterized for hyperbolic systems

Bounded spectrum implies cohomology to zero or constant

Results extended to continuous-time Anosov flows and geodesic flows

Abstract

In this paper, we study the structure of Birkhoff spectra for hyperbolic dynamical systems. Given a H\"older observable \(f\) on a basic set \(\Lambda\), 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 Liv\v{s}ic'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).