# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

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### 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).

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

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).

### Small scrambled sets

In a discrete dynamical system given by a metric space $X$ (with metric $d$) and a continuous map $f:X\to X$, a subset $S$ of $X$ containing at least two points is called a scrambled set if for any $x,y\in S$ with $x\ne y$,

In the lecture, systems with small scrambled sets will be constructed/studied. Possible cardinalities of maximal scrambled sets and the topological size of scrambled sets of minimal systems will be discussed. The results were obtained jointly with François Blanchard and Wen Huang.

### When is the attractor of a dissipative system in the cylinder homeomorphic to the circle?

We look for conditions under which the attractor of a dissipative system in the cylinder is homeomorphic to the circle. On the other hand, we show how the appearance of periodic inversely unstable solutions imply that the attractor is not homeomorphic to the circle. We apply these results to the particular case of the damped pendulum type equation driven by a periodic force.

### Symmetric functions and cylindric Schur functions

Although algebraic in nature, symmetric functions have long been of considerable interest in combinatorics. We will begin with a general introduction to the combinatorics of symmetric functions, where the two main highlights will be the Schur functions and the Littlewood-Richardson rule. Much of the appeal of Schur functions stems from their appearance in other areas of mathematics, and we will mention connections with representation theory, algebraic geometry (Schubert calculus, in particular) and matrix theory.
This will prepare us for the second half of the talk, where we will discuss cylindric Schur functions. As well as being a natural generalization of Schur functions, they are of much relevance to a fundamental open problem in algebraic combinatorics.
No knowledge of symmetric functions or combinatorics will be assumed.

### Subalgebras dense in Hardy spaces

Let $A$ be a subalgebra of ${H}^{\infty }$. For each $p>0$ the Hardy space in the unit disk ${H}^{p}\left(D\right)$ has a structure of $A$-module. The problem of describing the lattice of closed submodules of this module gives an example of the lattice of closed subspaces invariant under the action of a family of commuting operators. The investigation of the module structure of the disk-algebra was initiated by Wermer in 1950s. It leads to Beurling type results and includes the classical theorem of Beurling as a special case corresponding to the subalgebra of all polynomials. In particular, if the algebra $A$ is dense in all ${H}^{p}$, $p<\infty$, every closed submodule is $z$-invariant in a weak sense, and if $A$ is weak- $*$ dense in ${H}^{\infty }$, then every closed $A$-submodule of ${H}^{p}$ is $z$-invariant. In the talk we will discuss some recent results in this area.

### The Cauchy problem for nonlinear evolution equations with roughdata

The notions of local and global well-posedness for nonlinear evolution equations, such as Schrödinger- and generalized KdV-equations, are introduced and it is explained, how to apply the contraction mapping principle to obtain positive results concerning this question. The use of linear estimates, e.g. Strichartz-estimates, within this framework is discussed in view on lowering the regularity assumptions on the data, usually measured in terms of the classical Sobolev spaces ${H}^{s}$. The Fourier restriction norm method introduced by Bourgain in 1993 is sketched at hand of several applications. Finally we indicate some recent progress, which could be achieved by leaving the scale of the Sobolev spaces and applying an appropriate generalization of Bourgain's method.

### Extension of the Sharkovsky Theorem to non Autonomous Dynamical Systems I

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