14:30 - 14:55 Hamza Ounesli, ICTP

New manifolds supporting volume preserving homeomorphisms with exponential decay of correlations

Abstract: Manifolds known to support volume preserving homeomorphisms with exponential decay of correlation are all of the form N x K where N is up to a cover a hyperbolic infranilmanifold and K the unit tangent bundle of a manifold admitting a Riemannian metric with strictly negative sectional curvature. D. Dolgpyat and Y. Pesin, in the spirit of realization problems initiated by J. Von Neumann, asked if there are manifolds not supporting exponentially mixing systems with respect to a volume measure. Our main result, gives a class of systems that can be defined on any manifold of dimension at least 4 carrying a particular type of singular compact foliation, and exhibit exponential mixing with respect to the volume. Our second result building on the first result, shows that for any finitely presented group G. There exists a closed 4-manifold whose fundamental group is G and support the class of systems of the first result, this proves there are no topological restrictions at the homotopy level for existence of such systems.

15:05 - 15:30 Maik Gröger, Jagiellonian University

Besicovitch vs. Weyl mean equicontinuity

Abstract: The notions of Besicovitch and Weyl mean equicontinuity arise naturally in the study of systems with long-range order. In the minimal setting, it is well known that both describe the same class of actions: systems with discrete spectrum and continuous eigenfunctions. In this talk, I will show that this equivalence fails in the context of general actions by amenable but non-abelian groups, presenting explicit counterexamples involving the group of orientation-preserving homeomorphisms of the unit interval and the Lamplighter group. If time permits, I will describe some of the new tools developed to analyze these counterexamples.
This is joint work with G. Fuhrmann and T. Hauser.

16:00 - 16:25 Sonja Štimac, University of Zagreb

Classification of Hénon maps with strange attractors via the topology of a stable manifold

Abstract: In an earlier work with Jan Boroński, we classified (up to conjugacy) the Hénon maps with strange attractors using three invariants we introduced: (a) kneading sequences, (b) pruned trees, and (c) folding patterns of the unstable manifold of the hyperbolic fixed point in the attractor. In this talk, I will present another method for determining conjugacy classes of these maps, this time based on the topology of the stable manifold of the hyperbolic fixed point. We consider a region of dissipation for the Hénon map and study the connected components of the stable manifold within this region. To each such component, we assign a separation type and prove that two Hénon maps are conjugate if and only if their corresponding components share the same separation type. This is joint work with Jan Boroński.

16:35 - 17:00 Jan Boroński, Jagiellonian University

Topologically mildly dissipative homeomorphisms and Wang-Young Strange Attractors

Abstract: In this joint work with Sonja Štimac, we extend R.F. Williams’ result on 1-dimensional hyperbolic attractors to the non-uniformly hyperbolic setting, by showing that each Wang-Young strange attractor in the plane is conjugate to the shift on the inverse limit of a baobab (Peano continuum that contains at most one Jordan curve), generalizing our earlier result on Hénon attractors. More generally, the result holds on the core of the maximal attractor of any mildly dissipative diffeomorphism (in the sense of Crovisier and Pujals). We also generalize these results to the C0 setting, by introducing the class of topologically mildly dissipative surface homeomorphisms, providing a unified approach that covers many classes of dissipative dynamical systems scattered in the literature. Our purely topological conditions lead to a one-to-one correlation between the sets of ergodic measures of the one-dimensional and two-dimensional systems, as well as equality between the corresponding measure-theoretic entropies.