Dynamical Systems Seminars: 6 August at 16:30, ICTP LB Stasi Seminar Room

[ICTP Math Section]

*Dynamical Systems Seminars*

*Date and Venue*: Thursday 6 August at 16:30 - ICTP Leonardo Building, 
Stasi Seminar Room

*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.

All are welcome to join.