2015/01/15 by Gianluigi Del Magno, Del Magno, Gianluigi, João Lopes Dias +5
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1501.03697
openalex publication_date 2015/01/15 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We consider polygonal billiards with collisions contracting the reflection\nangle towards the normal to the boundary of the table. In previous work, we\nproved that such billiards has a finite number of ergodic SRB measures\nsupported on hyperbolic generalized attractors. Here we study the relation of\nthese measures with the ergodic absolutely continuous invariant probability\nmeasures (acips) of the slap map, the 1-dimensional map obtained from the\nbilliard map when the angle of reflection is always equal to zero. Our main\nresult states that for a generic polygon, if the reflection law has a Lipschitz\nconstant sufficiently small, then there exists a one-to-one correspondence\nbetween the ergodic SRB measures of the billiard map and the ergodic acips of\nthe corresponding slap map, and moreover that the number of Bernoulli\ncomponents of each ergodic SRB measure equals the number of the exact\ncomponents of the corresponding ergodic acip. The case of billiards in regular\npolygons and triangles is studied in detail.\n