# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

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### The Baum-Connes conjecture and linear group actions on spaces of finite asymptotic dimension

The Baum-Connes conjecture describes the $K$-theory of a reduced crossed product of a ${C}^{*}$-algebra by a group $G$ in terms of the $K$-homology of the classifying space of proper actions of $G$. We shall describe the Conjecture for discrete group case and its connotations. Commenting on results of Guentner, Higson, Kasparov, Weinberger, and Yu, we investigate the possibility of applying a finite-dimensionality argument in order to prove parts of the Conjecture for discrete linear groups.

### Hausdorff dimension for an open class of repellers in ${ℝ}^{2}$

We compute the Hausdorff dimension for an open class of Iterated Function Schemes in ${ℝ}^{2}$ in the ${C}^{2}$ topology, in terms of the pressure function. This class is characterized by a domination condition that is not too strong, plus the non-overlapping condition. Also in this class there exists a unique invariant probability measure of full Hausdorff dimension. Finally we show that the Hausdorff dimension is a continuous function in this class.

### Topology of minimal systems

I will start with a short survey of basic facts about minimal sets and I will briefly discuss their role in the theory of dynamical systems. In the second part, I will present recent results obtained with F. Balibrea, T. Downarowicz, L. Snoha and V. Spitalsky.

### Basic sets and distributional chaos in dimension one

The notion of basic set was introduced by A. N. Sharkovskii in the series of his papers from late sixties. These papers focused on omega-limit sets of continuous maps of the compact interval. It turned out later that existence of basic sets for a dynamical system on the compact interval corresponds to nontrivial behavior of the dynamical system: positive topological entropy, existence of horseshoes, distributional (and other) chaos.
In this talk we present results showing that similar behavior remains true when other one-dimensional spaces are considered (the circle, trees, graphs).

### Faithful linear representions of the free group

Many families of matrices that generate free groups are known, notably constructions by Hausdorff (1907) and Sanov (1947). Topological arguments show that those families are ubiquitous, and Tits' Theorem of Alternatives (1972) gives deep algebraic reasons for that. On the other hand, faced with a pair of matrices, it is almost hopeless to decide whether they generate a free group.
I will describe some that is known on explicit presentation of matrix generators of free groups, concentrating on joint work with Jairo Gonçalves and Mazi Shirvani, that describe explicit free groups in the multiplicative group of some finite dimensional algebras.

### Minimal realizations of Jewett-Krieger type for nonuniquely ergodic systems II

In a topological dynamical system $\left(X,T\right)$ the set of invariant measures is a Choquet simplex $K\left(X,T\right)$ whose extreme points are ergodic measures. We will prove that for any zero-dimensional system $\left(X,T\right)$ having no periodic points there exists a minimal system $\left(Y,S\right)$ and an affine homeomorphism $f$ between the Choquet simplexes $K\left(X,T\right)$ and $K\left(Y,S\right)$ such that for each $m$ in $K\left(X,T\right)$ the measure-theoretic systems $\left(X,m,T\right)$ and $\left(Y,f\left(m\right),S\right)$ are isomorphic. In this sense $\left(Y,S\right)$ is a minimal model for $\left(X,T\right)$. As an application we will derive the existence of minimal models with a preset collection of invariant measures (arranged in form of a simplex).

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