149 research outputs found
Billiard transformations of parallel flows: a periscope theorem
We consider the following problem: given two parallel and identically oriented bundles of light rays in R^{n+1} and given a diffeomorphism between the rays of the former bundle and the rays of the latter one, is it possible to realize this diffeomorphism by means of several mirror reflections? We prove that a 2-mirror realization is possible, if and only if the diffeomorphism is the gradient of a function. We further prove that any orientation reversing diffeomorphism of domains in R^2 is locally the composition of two gradient diffeomorphisms, and therefore can be realized by 4 mirror reflections of light rays in R^3, while an orientation preserving diffeomorphism can be realized by 6 reflections. In general, we prove that an (orientation reversing or preserving) diffeomorphism of wave fronts of two normal families of light rays in R^3 can be realized by 6 or 7 reflections
Large normally hyperbolic cylinders in a priori stable Hamiltonian systems
We prove the existence of normally hyperbolic invariant cylinders in nearly
integrable hamiltonian systems
Polymorphisms and adiabatic chaos
At the end of the last century Vershik introduced some dynamical systems, called
polymorphisms. Systems of this kind are multivalued self-maps of an interval, where
(roughly speaking) each branch has some probability. By definition, the standard Lebesgue
measure should be invariant. Unexpectedly, some class of polymorphisms appeared in
the problem of destruction of an adiabatic invariant after a multiple passage through a
separatrix. We discuss ergodic properties of polymorphisms from this class
- …