הסתברות ותורה ארגודית Ical Atom

A topological dynamical system (X,T) is said to be entropy minimal if all closed T-invariant subsets of X have entropy strictly less than (X,T). In this talk we will discuss the entropy minimality of a class of topological dynamical systems which appear as the space of graph homomorphisms from Z^d to graphs without four cycles; for instance, we will see why the space of 3-colourings of Z^d is entropy minimal even though it does not have any of the nice topological mixing properties.

``Topological structures‘‘ associated to a topological dynamical system are recently developed tools in topological dynamics. They have several applications, including the characterization of topological dynamical systems, computing automorphisms groups and even the pointwise convergence of some averages. In this talk I will discuss some recent developments of this subject, emphasizing applications to the pointwise convergence of some averages.

Using a result from Ergodic Ramsey theory, we answer a question posed by Bès, Martin, Peris and Shkarin by showing a mixing operator $T$ on a Hilbert space such that the tuple $(T, T^2)$ is not disjoint transitive.

We shall discuss the Chirikov standard map, an area-preserving map of the torus to itself in which quasi-periodic and chaotic dynamics are believed to coexist. We shall describe how the problem can be related to the spectral properties of a one-dimensional discrete Schroedinger operator, and present a recent result. Based on joint work with T. Spencer.

Property (T) was introduced by Kazhdan in 1967 as a way to establish compact generation of groups and since then was found useful for many other applications such as group cohomology and expander graphs. I will introduce a new notion of property (T) in Banach spaces, present a criterion for the fulfillment of this property and discuss its applications to the construction of Banach expanders and to group fix point properties in Banach spaces.

We prove that if the numerical range of a Hilbert space contraction $T$ is in a certain closed convex set of the unit disk which touches the unit circle only at 1, then $|T^n(I-T)| =\mathcal O(1/n^{\beta})$ with $\beta \in [\frac{1}{2}, 1)$. For normal contractions the condition is also necessary. Another sufficient condition for $\beta=\frac{1}{2}$, necessary for $T$ normal, is that the numerical range of $T$ be in a disk ${z: |z-\delta| \le 1-\delta}$ for some $\delta \in (0,1)$. As a consequence of results of Seifert, we obtain that a power-bounded $T$ on a Hilbert space satisfies $|T^n(I-T)| = \mathcal O(1/n^{\beta})$ with $\beta \in (0,1]$ if and only if $\sup_{1<|\lambda| <2} |\lambda -1|^{1/\beta} |R(\lambda,T)|< \infty$. When $T$ is a contraction on $L_2$ satisfying the numerical range condition, it is shown that $T^nf /n^{1-\beta}$ converges to 0 a.e. with a maximal inequality, for every $f \in L_2$. An example shows that in general a positive contraction $T$ on $L_2$ may have an $f \ge 0$ with $\limsup T^nf/\log n \sqrt{n} =\infty$ a.e.

We study the model of random walk in random environments in dimension four and higher under Sznitman‘s ballisticity condition (T‘). We prove a version of a local Central Limit Theorem for the model and also the existence of an equivalent measure which is invariant with respect to the point of view of the particle. This is a joint work with Noam Berger and Moran Cohen.