On the Birkhoff Spectrum for Hyperbolic Dynamics

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).