On the Birkhoff Spectrum for Hyperbolic Dynamics

In this paper, we study the structure of Birkhoff spectra for hyperbolic dynamical systems. Given a Hölder observable $f$ on a basic set $\mathrm{\Lambda }$, 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 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).