One-day workshop on Dynamical Systems

[Stefano Luzzatto]

One-day workshop on Dynamical Systems

Wednesday 11 January 2017

Venue: Luigi Stasi seminar room (Leonardo da Vinci Building) 10:30 - 11:30 Oliver Butterley (ICTP)

Open sets of exponentially mixing Anosov flows

Abstract: If a flow is sufficiently close to a volume-preserving Anosov flow and dim E_s = 1, dim E_u \geq 2 then the flow mixes exponentially whenever the stable and unstable foliations are not jointly integrable (similarly if the requirements on stable and unstable bundle are reversed). This implies the existence of non-empty open sets of exponentially mixing Anosov flows. (Joint work with Khadim War).

12:00 - 13:00 Davide Ravotti (Bristol)
Ergodic properties of area-preserving flows on compact surfaces

Abstract: We consider the set of area-preserving flows on compact surfaces with isolated fixed points. The study of these flows dates back to Novikov in the 80s and since then many properties have been investigated. Starting from an overview of the known results, we will show that typical such flows admitting several minimal components are mixing when restricted to each minimal component and we provide an estimate on the decay of correlations for smooth observables.

15:00 - 16:00 Maurizio Monge (UFRJ, Brazil)
Rigorous computation in random dynamics and computer-aided proof of noise-induced-order

Abstract: We will explain the techniques used to compute rigorously the invariant measure and rigorously estimate observables for general dynamical systems with noise. We will illustrate the interplay of Wasserstein, $L^1$ and variation norms on the space of signed measures, and show how it can be exploited to obtain a surprisingly effective estimation of the invariant measure in the $L^1$ norm. We conclude showing how this allows to prove rigorously the noise-induced order phenomenon for a model of the Belousov-Zhabotinsky reaction, that had been discovered with numerical simulations by Matsumoto-Tsuda in 1983.

16:30 - 17:30 Stefano Galatolo (Pisa)
A functional analytic approach for skew products with weakly contracting fibers. Application to the statistical stability

Abstract: We consider dynamical systems preserving a weaky contracting foliation. (Like Poincaré maps of Lorenz systems or toral extensions). We show some suitable spaces of measures with sign (whose regularity is defined trough a disintegration on the preserved foliation) adapted to this kind of systems. The spaces are well behaved with respect to the action of the transfer operator and in particular they satisfy a kind of Lasota Yorke inequality. By this, and by the speed of decay of correlations of these systems, which is known from previous works, we deduce quantitative statements about the statistical stability of such systems.

Everyone is welcome to attend 

****************************************************
Stefano Luzzatto
Mathematics Section
Abdus Salam International Centre for Theoretical Physics
Strada Costiera 11, 34151 Trieste, Italy
Tel +39 040 2240253
luzzatto at ictp.it
http://www.ictp.it/~luzzatto