BGU Probability and Ergodic Theory (PET) seminar Ical Atom

See attached file. This will be the first in a series of survey talks:

1. Operator ergodic theorems.

2. Ergodic and mixing theorems for Markov operators (discrete time Markov processes).

3. Ergodic theorems for random walks on locally compact groups (convolution powers). The second talk will focus on the results needed for the third one.

This is the second survey talk in the series. See attached file.

Let $x \to x+ \alpha$ be a rotation on the circle and let $\varphi$ be a function with bounded variation. Denote by $S_n(\varphi, x) := \sum_{j=0}^{n-1} \varphi(x+j \alpha)$ the ergodic sums.

For a large class of $\alpha$’s including irrationals with bounded partial quotients, we show decorrelation inequalities between the ergodic sums at time $q_k$, where the $q_k$’s are the denominators of $\alpha$.

This allows to study the asymptotic distribution of the ergodic sums $S_n(\varphi, x)$ after normalization, in particular for some step functions, along subsequences.

We will give also an application to a geometric model, the billiard flow in the plane with periodic rectangular obstacles when the flow is restricted to special directions.

See attached file

Random walks on lattices have been studied for decades and are by now very well understood. In this talk we will define a random walk on the primitive points of a lattice and discuss its properties. The random walk is obtained in a similar manner to the classical one with the difference that one divides by the gcd at each step. Subject to suitable conditions on the measure generating the walk, we will see how these random walks correspond to positive recurrent Markov chains. In particular we will see that there is a unique stationary distribution for these random walks.

In this talk we discuss the fine scale $L^2$-mass distribution of toral Laplace eigenfunctions with respect to random position. For the 2-dimensional torus, under certain flatness assumptions on the Fourier coefficients of the eigenfunctions and generic restrictions on energy levels, both the asymptotic shape of the variance and the limiting Gaussian law are established, in the optimal Planck-scale regime. We also discuss the 3-dimensional case, where the asymptotic behaviour of the variance is analysed in a more restrictive scenario. This is joint work with Igor Wigman.

We study correlation estimates of automatic sequences (that is, sequences computable by finite automata) with polynomial phases. As a consequence, we provide a new class of good weights for classical and polynomial ergodic theorems, not coming themselves from dynamical systems. We show that automatic sequences are good weights in L^2 for polynomial averages and totally ergodic systems. For totally balanced automatic sequences (i.e., sequences converging to zero in mean along arithmetic progressions) the pointwise weighted ergodic theorem in L^1 holds. Moreover, invertible automatic sequences are good weights for the pointwise polynomial ergodic theorem in L^r, r > 1. This talk is based on joint work with Tanja Eisner.