Après midi dynamique et spectre
Limit theorems for non-uniformly hyperbolic systems: a review.
This talk will focus on a family of non-uniformly hyperbolic dynamical systems, more specifically expading maps of the interval with a neutral fixed point (Pomeau-Manneville transformations). These transformations have an invariant measure with a pole at the neutral fixed point. I will describe some phenomena occuring with these maps (e.g. polynomial decay of correlations, invariant ergodic mesaure of infinite mass), and how they change as the fixed point becomes more or less neutral. There will be an emphasis on decay of correlations and distributional limit theorems, but, depending on the time, I may also present a few more recent and varied results (convergence of moments or variants of Li-Yorke chaos).
Exponential mixing for piecewise C^2 expanding semiflows.
The rate of mixing for continuous time hyperbolic systems is both a fundamental property of the system and crucial for more complicated systems, for example coupled models related to energy transport. In joint work with Peyman Eslami we extend results on exponential rates of mixing to some hyperbolic flows with discontinuities, moving closer to more physically relevant systems. I will explain the mechanism present in continuous time systems which produces the exponential mixing and describe the problems and solutions related to the low regularity of the systems.
Resonance-free zones for surfaces with cusps.
I will give a parametrix for the scattering determinant in the case of
surfaces with cusps of negative curvature. I will explain the
consequences for generic metrics, with different support assumptions. In
particular, I obtain many such surfaces with zones of non-constant
curvature for which the resonances of the Laplace operator are all in a
vertical strip. I will also discuss a specific example that exhibits a
sequence of resonances along a log line. If time allows, I will sketch
the way to obtain improved (in comparison with what is known) counting
estimates for resonances close to the axis.
For an exposition on what is known on the counting of resonances, and
references on surfaces with cusps, see http://arxiv.org/abs/1405.5445