16:30-17:15 Rubio Gunawan (SISSA)
Smooth Circle Covering with a Physical Measure on a Hyperbolic Repelling Fixed Point

Abstract:
We construct an example of a smooth (C-infinity) circle covering map topologically conjugate to the doubling map, such that it has a physical measure supported on a hyperbolic repelling fixed point. By relaxing the smooth condition at a single point, we also construct an example where the basin of the physical measure has full measure. A key technical step is a realization method of independent interest, which gives a canonical way to construct a full branch map given its induced map.

17:30-18:15 Sergio Romana (Sun Yat-sen University, China)
Birkhoff Spectra for Hyperbolic Dynamics: Density, Rigidity, and Geometric Applications

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.