## Quantitative multiple recurrence for two and three transformations.

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...

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

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...

Read More## The group of reversible Turing machines and the torsion problem for $\Aut(A^{\mathbb{Z})$ and related topological fullgroups.

Abstract: We introduce the group $RTM(G,n,k)$ composed of abstract Turing machines which use the group $G$ as a tape, use an alphabet of $n$ symbols, $k$ states and act as a bijection on the set of configurations. These objects can be represented both as cellular automata and in terms of continous functions and cocycles. The study of this group structure yields interesting results concerning computability properties of some well studied groups such as $\Aut(A^{\mathbb{Z})$ and the topological full group of the two dimensional full shift. More precisely, given a finitely generated group $G...

Read More## New techniques for pressure approximation in Z^d shift spaces

ABSTRACT: Given a Z^d shift of finite type and a nearest-neighbour interaction, we present sufficient conditions for efficient approximation of pressure and, in particular, topological entropy. Among these conditions, we introduce a combinatorial analog of the measure-theoretic property of Gibbs measures known as strong spatial mixing. The approximation techniques make use of a special representation theorem for pressure that may be of independent interest. Part of this talk is joint work with Stefan Adams, Brian Marcus, and Ronnie...

Read More## Piecewise contraction maps and its applications

Abstract: Our talk concerns dynamical systems which are defined by piecewise contraction maps (PC maps). There is a large literature which deals with the dynamical behavior of PC maps defined on convex subsets of Euclidean spaces in different contexts. Our aim is to show that, under certain conditions, a typical PC map, in the measure theoretical sense of the parameter space, is asymptotically periodic which means that the map has finitely many periodic orbits and every orbit converges to a periodic orbit. Our setup is the following: We fix an Iterated Function System {φ1, . . . ,...

Read More## Limit theorems for products of non-negative and tropical random matrices

Abstract: Tropical matrices are matrices with entries in $(\R\cup\{-\infty\}, \max,+)$, where $\max$ is seen as the “addition”. They appear as the limit of nonnegative matrices and in models from computer science/operations research, as they are strongly linked to weighted directed graphs. Their dynamical behavior is quite similar to nonnegative matrices, with a kind of Perron-Frobenius theorem but the limits are often reached, which allows some combinatorial studies. In this talk, I will present a common framework, known as topical maps, to deal with both nonnegative matrices...

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