2012/03/06 by Vadim Kaloshin, Kaloshin, Vadim, Alfonso Sorrentino +1 · 1 citation
Mathematics · Physics and Astronomy · #37D50 #37E40 #37J35 #37J50 #53C24 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #Symplectic Geometry (math.SG) #math.DS #math.SG #msc:37D50 #msc:37E40 #msc:37J35 #msc:37J50 #msc:53C24
paper · pdf · doi:10.48550/arxiv.1203.1274
23 pages, 3 figures
arxiv created 2012/03/06 · openalex publication_date 2012/03/06 · arxiv updated 2012/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a strictly convex domain Ω in \R2, there is a natural way to define a billiard map in it: a rectilinear path hitting the boundary reflects so that the angle of reflection is equal to the angle of incidence. In this paper we answer a relatively old question of Guillemin. We show that if two billiard maps are C1,α-conjugate near the boundary, for some α> 1/2, then the corresponding domains are similar, i.e. they can be obtained one from the other by a rescaling and an isometry. As an application, we prove a conditional version of Birkhoff conjecture on the integrability of planar billiards and show that the original conjecture is equivalent to what we call an "Extension problem". Quite interestingly, our result and a positive solution to this extension problem would provide an answer to a closely related question in spectral theory: if the marked length spectra of two domains are the same, is it true that they are isometric?