Topics

Andrzej Biś, University of Lodz, Poland
Variational principles for finitely generated pseudogroup actions

We study the thermodynamical formalism of finitely generated pseudogroups acting on a compact metric spaces. More precisely, we introduce the notions of topological and measure-theoretical pressures for those actions and we prove variational principle. The talk is based on joint work with Maria Carvalho and Paulo Varandas.

Melih Emin Can, Jagiellonian University, Poland
On asymptotic average shadowing property and vague specification property

Łukasz Kotlewski, Nicolaus Copernicus University, Poland
Conformal measures for interval exchange transformations

The aim of this talk is to present the topic of conformal measures in the class of dynamical systems called „interval exchange transformations”. Conformal measure describes the distribution of a phase space which evolves with time exponentially, with respect to piecewise potential. I will recall basic notions of interval exchange transformations and after that, I will present results from my master’s thesis.

Abdul Gaffar Khan, Nicolaus Copernicus University, Poland
Topological stability and persistent property from Gromov-Hausdorff Viewpoint

In this talk, we introduce topologically IGH-stable, IGH-persistent and pointwise weakly topologically IGH-stable homeomorphisms of compact metric spaces. We will prove that every topologically IGH-stable homeomorphism is topologically stable as well as topologically GH-stable and every expansive topologically stable homeomorphism of a compact manifold is topologically IGH-stable. We further prove that every equicontinuous pointwise weakly topologically IGH-stable homeomorphism is IGH-persistent and every pointwise minimally expansive IGH-persistent homeomorphism is pointwise weakly topologically IGH-stable. These results strengthen the well known Walters’ stability theorem from the viewpoint of pointwise dynamics as well establish the relation between persistent properties and topological stability from the Gromov-Hausdorff viewpoint. This talk is based on joint work with C.A. Morales and Tarun Das.

Elias Rego, AGH, Poland
On the Ergodic Theory of ASH Attractors

In this talk we will recall some weak notions of hyperbolicity for flows. The Lorenz and the Rovella attractors are the most famous examples of such theory. We will present new results towards the ergodic theory of ASH attractors. This is a joint work with A. Arbieto, M. Pineda and K. Vivas.

Sophie Schmidhuber, University of Zurich, Switzerland
Algorithmic decomposition of generalized interval exchange maps

In this talk, we present an algorithm which extends the classical Rauzy-Veech induction in order to study generalized interval exchange transformation (GIETs) with possibly more than one quasiminimal component (i.e. not infinite-complete, or, equivalently, not semi-conjugated to a minimal IET). The algorithm is defined for more general maps that we call interval exchange transformations with gaps (g-GIETs), namely partially defined GIETs which appear naturally as the first return map of Cr -foliations on two-dimensional manifolds to any transversal segment. We exploit the algorithm to find a decomposition of any GIET into a finite unions of intervals which either contain no recurrent orbit, or contain only recurrent orbits which are closed, or contain a unique quasiminimal. This provides an alternative algorithmic approach to the decomposition results for foliations and flows on surfaces by Levitt, Gutierrez and Gardiner from the 1980s. This is joint work with Corinna Ulcigrai and Charles Fougeron.

Alexandre Trilles, Jagiellonian University, Poland
Loose Bernoullicity for zero entropy flows via Feldman-Katok pseudometric

Our goal is to study measure-preserving flows consisting of a continuous flow on a compact metric space and an ergodic measure with zero entropy. More precisely, we will focus on loosely Bernoulli flows with zero entropy, which, following Ratner's suggestion, are called loosely Kronecker.

The Feldman-Katok pseudometric (FK-pseudometric), originally defined for transformations, will be introduced in the context of flows and used to recover the results of García-Ramos and Kwietniak, characterizing loosely Kronecker systems. We will show that a measure-preserving flow is loosely Kronecker if and only if there exists a full-measure set where any pair of points is indistinguishable with respect to the FK-pseudometric. Additionally, a purely topological characterization of uniquely ergodic flows whose unique invariant measure is loosely Kronecker will be provided."

Yuriy Tumarkin, University of Zurich, Switzerland
Invariant measures for periodic type skew-products over IETs

The problem of understanding the invariant measures for the linear flow on compact translation surfaces (and correspondingly for interval exchange transformations) has been much studied since the 1970s and is by now very well understood. Very little is however known about infinite translation surfaces. I will talk about the simplest case of infinite translation surfaces, which are obtained as $\Z^m$-covers of compact translation surfaces, giving rise to skew-products over IETs. It turns out that in the case when the skew-product is periodic under renormalisation one can classify all the ergodic invariant measures and even obtain a somewhat explicit form for them.