Billiard and diophantine approximation
We consider the billiard map in the hypercube of . 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 is the order of magnitude of the complexity.
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.
Binary operations on algebras of observables are studied in the quantum as well as in the classical case. It is shown that certain natural compatibility conditions with the associative product imply properties which are usually additionally required.
This paper is devoted to the study of traveling waves for monotone evolution systems of bistable type. In an abstract setting, we establish the existence of traveling waves for discrete and continuous-time monotone semiflows in homogeneous and periodic habitats. The results are then extended to monotone semiflows with weak compactness. We also apply the theory to four classes of evolution systems.
It is proved for Abelian groups that the Reidemeister coincidence number of two endomorphisms ϕ and ψ is equal to the number of coincidence points of ϕ̂ and ψ̂ on the unitary dual, if the Reidemeister number is finite. An affirmative answer to the bitwisted Dehn conjugacy problem for almost polycyclic groups is obtained. Finally, we explain why the Reidemeister numbers are always infinite for injective endomorphisms of Baumslag-Solitar groups.
By using the ideas introduced by McGehee in the study of the singularities in some problems of Celestial Mechanics, we study the singularities at the origin and at the infinity for some classical mechanical systems with homogeneous kinetic and potential energy functions. For these systems the origin and the infinity of the configuration coordinates is usually a singularity or a nullity of the Hamiltonian function and the verctor field. This work generalizes a previous one by the first and the third...
We show that strongly positively recurrent Markov shifts (including shifts of finite type) are classified up to Borel conjugacy by their entropy, period and their numbers of periodic points.