arXiv subjects

Serge Troubetzkoy

Publications and source records attributed to Serge Troubetzkoy.

57 records · Page 4Linked to original sources

arXiv1994

Measures with infinite Lyapunov exponents for the periodic Lorentz gas

In \cite{Ch91a} it was shown that the billiard ball map for the periodic Lorentz gas has infinite topological entropy. In this article we study the set of points with infinite Lyapunov exponents. Using the cell structure developed in \cite{BSC90,Ku} we construct an ergodic invariant probability measure with infinite topological entropy supported on this set. Since the topological entropy is infinite this is a measure of maximal entropy. From the construction it is clear that there many such measures can coexist on a single component of topological transitivity. We also construct an ergodic invariant probability measure with finite entropy which is supported on this set showing that infinite exponents do not necessarily lead to infinite entropy.

math.DS↗

arXiv1994

Some remarks on periodic billiard orbits in rational polygons

A polygon is called rational if the angle between each pair of sides is a rational multiple of $π.$ The main theorem we will prove is Theorem 1: For rational polygons, periodic points of the billiard flow are dense in the phase space of the billiard flow. This is a strengthening of Masur's theorem, who has shown that any rational polygon has ``many'' periodic billiard trajectories; more precisely, the set of directions of the periodic trajectories are dense in the set of velocity directions $§^1.$ We will also prove some refinements of Theorem 1: the ``well distribution'' of periodic orbits in the polygon and the residuality of the points $q \in Q$ with a dense set of periodic directions.

math.DS↗

arXiv1994

Acceleration of bouncing balls in external fields

We introduce two models, the Fermi-Ulam model in an external field and a one dimensional system of bouncing balls in an external field above a periodically oscillating plate. For both models we investigate the possibility of unbounded motion. In a special case the two models are equivalent.

math.DS↗