Theorem of Livšic type for dispersed billiards (Q1804137)

The author investigates a dynamical system generated by the motion of a point particle in a closed region \(M\) which is obtained from a two-dimensional torus by cutting out a finite number of pairwise nonintersecting convex regions. The author assumes that each of these regions is bounded by a convex curve \(\Gamma_j\), \(j=1, \dots, \ell\) of class \(C^3\) and the particle moves rectilinearly with unit velocity within \(M\) and the point is reflected from the boundary \(\partial M\) in accordance with the law of elastic impact. Piecewise-Hölder functions on the phase space of a dispersed billiard are considered. It is shown that if the integral of such a function around any periodic trajectory is zero then the function itself is cohomologous to zero.