We discuss joint work with Douglas Arnold, Guy David, Marcel Filoche and Svitlana Mayboroda. Consider for the operator $L = -\Delta + V$ with periodic boundary conditions, and more generally on the manifold with or without boundary. Anderson localization, a significant feature of semiconductor physics, says that the eigenfunctions of $L$ are exponentially localized with high probability for many classes of random potentials $V$. Filoche and Mayboroda introduced the function $u$ solving $Lu = 1$ and showed numerically that it strongly reflects this localization.

Abstract. We consider families of holomorphic maps defined on subsets of the complex plane,

_

Abstract: Paul L\'evy's classical arcsine law states that the occupation time ratio of one-dimensional Brownian motion for the positive side is arcsine-distributed. The arcsine law has been generalized to a variety of classes of stochastic processes and dynamical systems.

Abstract. The realization of m.p automorphisms as transfer on the space of the paths on the graded graphs allows to use new kind of encoding of one-sided Bernoulli shift. I will start with simple example how to realize Bernoulli shift in the locally finite space (graph) $\prod_n {1,2,\dots n}$ (triangle compact.) Much more complicated example connected to old papers by S.Kerov-Vershik and recent by Romik-Sniady in which one-sided Bernoulli shift is realized as Schutzenberger transfer on the space of infinite Young tableaux with Plancherel Measure. These examples open series of

Singular vectors are the ones for which Dirichlet’s theorem can be infinitely improved. For example, any rational vector is singular. The sequence of approximations for any rational vector q is 'obvious'; the tail of this sequence contains only q. In dimension one, the rational numbers are the only singulars. However, in higher dimensions there are additional singular vectors. By Dani's correspondence, the singular vectors are related to divergent trajectories in Homogeneous dynamical systems. A corresponding 'obvious' divergent trajectories can also be defined.

Abstract: We first survey a recent progress related to the nonsingular Bernoulli transformations. Then we construct inductively new examples of conservative Bernoulli maps of type III. They appear as a limit of a sequence of Bernoulli maps of type II_1.

Poisson suspensions are random sets of points endowed with a transformation that displaces each point according to a single transformation of the sigma-finite space where the points lie. In this ongoing work, instead of dealing with measure-preserving transformations (which is the classical case), we are going to present our attempt to explore the non-singular case. The difficulties are counterbalanced by new tools that are trivial in the measure-preserving case but highly informative in the non-singular one. We will present these tools as well as the first basic results we’ve obtained.

Abstract:

Abstract: We show that under certain conditions, a random walk on the 1-dim torus by affine expanding maps has a unique stationary measure. We then use this result to show that given an IFS of contracting similarity maps of the real line with a uniform contraction ratio 1/D, where D is some integer > 1, under some suitable condition, almost every point in the attractor of the given IFS (w.r.t. a natural measure) is normal to base D.

Abstract: A collection of polygons with the property that to each side one can find another side parallel to it can be endowed with a translation surface structure by glueing along those edges. This means that the closed surfaces obtained carries a flat metric outside finitely many conical singularities. Geodesics (which are straight lines) connecting such singularities are called saddle connections.

A powerful theory due to Ornstein and his collaborators has been successfully applied to many random systems to show that they are isomorphic to independent and identically distributed systems. The isomorphisms provided by Ornstein's theory may not be finitary, that is, effectively realizable in practice. Despite the large number of systems known to be Bernoulli, there are only a handful of cases where explicit finitary isomorphisms have been constructed. In this talk, we will discuss classical and recent constructions, and some long standing open problems.

Mean dimension is a topological invariant of dynamical systems introduced by Gromov that measures the number of parameters per iteration needed to describe a trajectory in the system. We characterize this invariant (at least for dynamical systems with the marker property, such as infinite minimal systems) using a min-max principle, where choices of both a metric on the topological space and an invariant probability measure on the system are varied. The work I will report on is joint work with M. Tsukamoto.