הסתברות ותורה ארגודית Ical Atom

Remarkably, understanding the behaviour of approximate groups allows us to convert information $A^k$ for some single fixed k into information about the sequence $A^n$ as n tends to infinity, and in particular about the growth of A.

In this talk I will provide some counter-examples for quantitative multiple recurrence problems for systems with more than one transformation. For instance, I will show that there exists an ergodic system $(X,\mathcal{X},\mu,T_1,T_2)$ with two commuting transformations such that for every $\ell < 4$ there exists $A\in \mathcal{X}$ such that $\mu(A\cap T_1^n A\cap T_2^n A) < \mu(A)^{\ell}$ for every $n \in \mathbb{N}$. The construction of such a system is based on the study of ``big‘‘ subsets of $\mathbb{N}^2$ and $\mathbb{N}^3$ satisfying combinatorial properties.

This a joint work with Wenbo Sun.

We propose a procedure (the first of its kind) for computing a fully data-dependent interval that traps the mixing time t_mix of a finite reversible ergodic Markov chain at a prescribed confidence level. The interval is computed from a single finite-length sample path from the Markov chain, and does not require the knowledge of any parameters of the chain. This stands in contrast to previous approaches, which either only provide point estimates, or require a reset mechanism, or additional prior knowledge.

The interval is constructed around the relaxation time t_relax, which is strongly related to the mixing time, and the width of the interval converges to zero roughly at a sqrt{n} rate, where n is the length of the sample path. Upper and lower bounds are given on the number of samples required to achieve constant-factor multiplicative accuracy. The lower bounds indicate that, unless further restrictions are placed on the chain, no procedure can achieve this accuracy level before seeing each state at least \Omega(t_relax) times on the average. Finally, future directions of research are identified.

Join work with Daniel Hsu and Csaba Szepesvári

We will discuss an equivalence between the existence of weak nets and the Radon number, in an abstractly convex space. Based on work with Shay Moran.

In 1969, H. Furstenberg proposed a conjecture on dimension of intersections of $\times 2$ and $\times 3$ -invariant sets. We will present some of the steps involved in a recent solution of this conjecture.

Devaney’s definition of chaos is one of the most popular and widely known. A continuous map f from a compact metric space (X, d) to itself is chaotic in the sense of Devaney if (1) f is transitive, (2) the set of all periodic points of f is dense in X, and (3) f has sensitive dependence on initial conditions. we will show that (1) and (2) imply (3) in Devaney’s definition in any metric space. We will show (1) and (3) do not imply (2) and (2) ; (3) do not imply (1). We will also show that for continuous maps on an interval in R; transitivity implies that the set of periodic points is dense. It follows that transitivity implies chaos, and we will give some examples to note that there are no other trivialities in Devaney’s definition when restricted to interval.