Typicality of finite complexity in planar dispersing billiards

13.01.2023 15:15 - 16:45

Peter Imre Toth (BME Budapest)

Abstract: In a hyperbolic dynamical system (map) T with singularities, the "complexity" of the singularity structure K(n) is the number of smoothness components of the n-th iterate T^n that can meet at a single phase point. Having an upper bound on K(n) plays an important role in many arguments about statistical properties, including Young tower constructions in dispersing billiards. In this talk I show that K(n) is bounded (by 4) for C^3-typical planar dispersing billiards with finite horizon and no corner points. This result has been unpublished since 2006, because it had no applications: it seemed that the much easier linear bound is enough for everything, and the much more interesting multi-dimensional generalization is hopeless. However, it turned out recently that there are people interested in the finite complexity case: new results of Baladi, Carrand, Demers and Korepanov on the measure of maximal entropy require stronger than linear bounds, and my result implies the typicality of their conditions.


Henk Bruin, Peter Balint


IST Austria, Big Seminar room Ground floor / Office Bldg West (I21.EG.101)