Working Group 2021-22 - Monday 14:00-16:00 - 16-26/209

September 20, 2021-January 24, 2022


Sep 20
Roberto CASTORRINI
Title: A thermodynamic estimate for the essential spectral radius: the piecewise expanding case
Abstract: We study the transfer operator associated to a piecewise expanding map on a compact manifold in finite dimension, acting on suitable Sobolev spaces. The aim is to prove a new estimate for the essential spectral radius in terms of thermodynamics quantities, such as the topological pressure.

Oct 4
Caroline WORMELL
Title: Linear response for the Lozi map and mixing of SRB measure cross-sections
Abstract: Dynamical systems of high dimension are expected, regardless of their hyperbolicity properties, to have a linear response, but this property does not follow from the existing theory. In a non-rigorous study, Ruelle proposed that linear response arose from generic average dynamics of singularities in the SRB measure with a large enough stable dimension: we pursue this idea in the setting of 2D piecewise hyperbolic maps, whose equivalents in one dimension fail to have linear response.
We show, rigorously, that linear response formally obtains for these maps if the SRB measure has a certain regenerating property: that its conditional measure on the singularity set - a measure supported on a set of dimension strictly less than one - converges exponentially back to the SRB measure under being pushed forward. We conjecture that this property holds generically, presenting strong numerical evidence and some parallels with some recent one-dimensional results. This unexpected phenomenon may furnish a general mechanism for linear response in higher dimensions, as well as being of its own interest.

Oct 11
Caroline WORMELL
Title: Linear response for the Lozi map and mixing of SRB measure cross-sections/Part II

Oct 18
Caroline WORMELL
Title: Linear response for the Lozi map and mixing of SRB measure cross-sections/Part III
Roberto CASTORRINI
Title: A thermodynamic estimate for the essential spectral radius: the piecewise expanding case/Part II

Oct 25
Roberto CASTORRINI
Title: A thermodynamic estimate for the essential spectral radius: the piecewise expanding case/Part III

Nov 8
NO SESSION (WORKSHOP IN NANTES)

Nov 15
Alexey KOREPANOV
Title: A natural symbolic model with polynomially mixing MME
Abstract: I'll present a convoluted proof that a quintupling interval map is polynomially mixing with respect to the measure of maximal entropy. A much stronger result of exponential mixing is also much simpler, but I restrict to methods which have a chance of working for dispersing billiards.

Nov 22
Alexey KOREPANOV
Title: A natural symbolic model with polynomially mixing MME, Part II

Nov 29
Jérôme CARRAND
Title: Existence of MME for the Sinai billiard flow
Abstract: Using Abramov's formula, we relate the MME of the Sinai billiard flow to the equilibrium states under a specific potential of the collision map. We give two conditions such that any H\"older potential satisfying them admits equilibrium states. Existence of equilibrium measures is proved by constructing such a measure from the maximal eigenvectors of an appropriate transfer operator acting on an anisotropic Banach space.

Dec 6
Julien SEDRO
Title: Quenched limit theorems for expanding on average cocycles.
Abstract: I will present joint works with Davor Dragicevic and Yeor Hafouta, where we extend the Nagaev-Guivarc'h spectral method, initially devised to prove probabilistic limit theorems for processes generated by deterministic hyperbolic dynamical systems, to the 'quenched' random case, where we look at processes generated by cocycles that are only expanding on average. Compared to the case where one composes at random uniformly expanding systems, which was already known, the expanding on average presents the additional difficulty of non-uniformity in the decay of random correlations. We take care of this by looking at renormalized observables, and by introducing a tool called adapted norms. This allow us to obtain various limit theorems: central limit theorem, local limit theorem, large deviations principle, almost-sure invariance principle...

Dec 13
No session (Aussois)


Jan 17
No Session (Autrans)

Jan 24
Julien SEDRO
Title: Fractional susceptibility functions for the quadratic family.
Abstract : In the last two decades, a clear picture of linear response for one-dimensional dynamics emerged. However, in the specific instance of the quadratic family, two seemingly contradictory results coexist: Ruelle proved in 2005, through the study of the susceptibility function, that the formal candidate for the derivative of the response function at 0 was well defined, while Baladi et al. proved in 2014 that the same response function was at best 1/2 Hölder at 0. To try to "shed some light on this puzzling state of affairs", Baladi and Smania introduced so called fractional susceptibility functions, which are complex functions of two complex variables, and relate to fractional response in the same way the susceptibility function relates to linear response. I will present results establishing holomorphy on a disk of radius larger than one for simplified fractional susceptibility functions, as well as explain the strategy to extend this result to the 'true' fractional susceptibility function.