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Ballistic regime for random walks in random environment with unbounded jumps and Knudsen billiards

Francis Comets, Serguei Popov (2012)

Annales de l'I.H.P. Probabilités et statistiques

We consider a random walk in a stationary ergodic environment in ℤ , with unbounded jumps. In addition to uniform ellipticity and a bound on the tails of the possible jumps, we assume a condition of strong transience to the right which implies that there are no “traps.” We prove the law of large numbers with positive speed, as well as the ergodicity of the environment seen from the particle. Then, we consider Knudsen stochastic billiard with a drift in a random tube in ℝ d , d ≥ 3 , which serves as environment....

Bifurcation of periodic and chaotic solutions in discontinuous systems

Michal Fečkan (1998)

Archivum Mathematicum

Chaos generated by the existence of Smale horseshoe is the well-known phenomenon in the theory of dynamical systems. The Poincaré-Andronov-Melnikov periodic and subharmonic bifurcations are also classical results in this theory. The purpose of this note is to extend those results to ordinary differential equations with multivalued perturbations. We present several examples based on our recent achievements in this direction. Singularly perturbed problems are studied as well. Applications are given...

Billiard complexity in the hypercube

Nicolas Bedaride, Pascal Hubert (2007)

Annales de l’institut Fourier

We consider the billiard map in the hypercube of ℝ d . We obtain a language by coding the billiard map by the faces of the hypercube. We investigate the complexity function of this language. We prove that n 3 d - 3 is the order of magnitude of the complexity.

Billiards and boundary traces of eigenfunctions

Steve Zelditch (2003)

Journées équations aux dérivées partielles

This is a report on recent results with A. Hassell on quantum ergodicity of boundary traces of eigenfunctions on domains with ergodic billiards, and of work in progress with Hassell and Sogge on norms of boundary traces. Related work by Burq, Grieser and Smith-Sogge is also discussed.

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