Abstract:
This lecture presents a systematic study of Birkhoff spectra for hyperbolic dynamical systems, focusing on the set of Birkhoff sums along periodic orbits. We begin by characterizing when the Birkhoff spectrum of a Hölder continuous observable on a basic set is dense in the real line, showing that dispersion and non-arithmeticity are key conditions. Our first main rigidity result states that boundedness of the spectrum forces the observable to be cohomologous to zero—extending the classical Livsic theorem. We further prove that if the spectrum is contained in a finite union of arithmetic progressions, the observable must be arithmetic, and for Anosov diffeomorphisms, cohomologous to a constant.
These results are then extended to continuous-time systems, including Anosov flows and geodesic flows on Anosov manifolds. In particular, we generalize a theorem of Dairbekov and Sharafutdinov by showing that boundedness of the closed geodesic spectrum forces a smooth function to vanish identically, while arithmetic sparseness forces it to be constant. Applications include rigidity results for Ricci curvature and Lyapunov exponents, illustrating how dynamical spectral constraints encode deep geometric information. The talk concludes with a discussion of the underlying techniques: shadowing, homoclinic intersections, and number-theoretic lemmas on rational independence and upper density.
