The seminar meets on Tuesdays, 10:50-12:00, in Math -101

2015–16–A meetings

Date
Title
Speaker
Abstract
Oct 27 Entropy Minimality and Four-Cycle Free Graphs Nishant Chandgotia (Tel Aviv)

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.

Nov 3 Topological structures and the pointwise convergence of some multiple averages for commuting transformations Sebastián Donoso (Universidad de Chile/ Hebrew University)

``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.

Nov 10 A mixing operator T for which (T, T^2) is not disjoint transitive Yunied Puig de Dios (BGU)

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.

Dec 8 Applications of discrete Schroedinger equations to the standard map Mira Shamis (Weizmann)

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.

Dec 15 Strengthening of Banach property (T) and applications Izhar Oppenheim (BGU)

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.

Dec 22 REMARKS ON RATES OF CONVERGENCE OF POWERS OF CONTRACTIONS Guy Cohen (BGU)

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.

Dec 29 Local limit theorem for strongly ballistic random walk in random environments Ron Rosenthal (ETH Zurich)

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.

Jan 5 Relative complexity of random walks in random sceneries in the absence of a weak invariance principle for local times Zemer Kosloff (Warwick)