# Dynamical Systems

## Quantitative multiple recurrence for two and three transformations.

Event Date: Apr 10, 2017 in Dynamical Systems, Seminars

Abstract:  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...

## TOPOLOGICAL DYNAMICS OF PIECEWISE Λ-AFFINE MAPS OF THE INTERVAL

Event Date: Apr 17, 2017 in Dynamical Systems, Seminars

ABSTRACT: Let 0 < a < 1, 0 ≤ b < 1 and I = [0,1). We call contracted rotation the interval map φa,b : x ∈ I → ax+b mod1. Once a is fixed, we are interested in the dynamics of the one-parameter family φa,b, where b runs on the interval interval [0, 1). Any contracted rotation has a rotation number ρa,b which describes the asymptotic behavior of φa,b. In the first part of the talk, we analyze the numerical relation between the parameters a, b and ρa,b and discuss some applications of the map φa,b. Next, we introduce a generalization of contracted rotations. Let −1 < λ < 1 and f...