Wandering set

Abstract: In the present paper we introduce the notion of dilation of a multiparametric linear stationary dynamical system (systems of this type, in particular dissipative, and conservative scattering ones were first introduced in [6]). We establish the criterion for existence of a conservative dilation of a multiparametric dissipative scattering system. This allows to distinguish the class of so-calledN-dissipative systems preserving the most important properties of one-parametric dissipative scattering systems.

Abstract: A {\em singular hyperbolic attractor} for flows is a partially hyperbolic attractor with singularities (hyperbolic ones) and volume expanding central direction \cite{mpp1}. The geometric Lorenz attractor \cite{gw} is an example of a singular hyperbolic attractor. In this paper we study the perturbations of singular hyperbolic attractors for three-dimensional flows. It is proved that any attractor obtained from such perturbations contains a singularity. So, there is an upper bound for the number of attractors obtained from such perturbations. Furthermore, every three-dimensional flow $C^r$ close to one exhibiting a singular hyperbolic attractor has a singularity non isolated in the non wandering set. We also give sufficient conditions for a singularity of a three-dimensional flow to be stably non isolated in the nonwandering set. These results generalize well known properties of the Lorenz attractor.

Abstract: We introduce a class of infinite-dimensional diffusion processes which contains a limiting version of the Ohta-Kimura model in population genetics. For this a necessary and sufficient condition for existence of stationary distributions is obtained. We are especially interested in the case where there is no stationary distribution. Then it is shown that an individual ergodic theorem holds for a suitably centralized process. As a corollary the wandering distribution exists.

Abstract: We investigate in which cases the boundary of a multiply connected wandering domain of an entire function is uniformly perfect. We give a general criterion implying that it is not uniformly perfect. This criterion applies in particular to examples of multiply connected wandering domains given by Baker. We also provide examples of infinitely connected wandering domains whose boundary is uniformly perfect.