Dynamics Day, 23 July - Stasi Room at 14:30

[ICTP Math Section]

*TOMORROW - 23 July, 14:30 - 17:00*
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*Stasi Room, ICTP Leonardo Building*
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*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.

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*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.

All are welcome to join.

-- 
Micol Stock
Secretariat of the ICTP Math Section

Abdus Salam International Centre for Theoretical Physics
Strada Costiera 11
34151 Trieste
Italy

Tel.: (+39) 040 2240 4455