Bounded Variation Spaces Research Articles

A spectral gap for transfer operators of piecewise expanding maps

We consider piecewise $\C^{1+\alpha}$ uniformly expanding maps ona Riemannian manifold, and study their invariant physical measures.We study the Perron-Frobenius operator on Sobolev spaces and boundedvariation spaces, and prove that it is quasicompact if someconditions on the Lyapunov exponent and the combinatorialcomplexities are satisfied. Then, we get strong results concerningthe existence of physical ergodic measures, and the exponentialmixing of smooth observables.

New Results on the Vibrating String with a Continuous Obstacle

We give an explicit formula which describes the solution of the problem of the linear elastic string vibrating against a plane obstacle without loss of energy. This formula allows us to prove continuous dependence on the initial data; a regularity result in some bounded variation spaces is given. A numerical scheme is deduced from the explicit formula. Finally we prove the weak convergence of a subsequence of solutions of the penalized problem to a “weak” solution (i.e. one which does not necessarily conserve energy) of the problem with an obstacle when the obstacle is arbitrary; when the obstacle is plane, all the sequence strongly converges to the solution of the obstacle problem which conserves the energy.