Special Issue: Statistical Properties of Dynamical Systems
Résumé
The study of statistical properties of dynamical systems is a very active direction in modern
ergodic theory. Roughly speaking, we identify two main themes: the first is to look for
probabilistic features of deterministic dynamical systems; the second is to make the system
itself random by adding noise to the system. The majority of the papers in this issue address
the first of these themes.
The papers by Freitas and Haydn deal with a quantitative study of recurrence for mixing
measures, by looking respectively at the extreme values statistics and at the statistics of
successive and multiple returns in small sets: in both cases one gets precise estimates on the
occurrence of rare events. Recurrence reflects the dynamical properties of ergodic measures;
their geometrical properties are instead captured by multifractal analysis. Iommi and Todd
describe the multifractal structure and the thermodynamic formalism of multimodal maps
by looking at inducing schemes for such maps. The article by Bruin and Van Strien explores
the detailed structure, in the category of multimodal interval maps, of the level sets of
constant topological entropy. Perturbations of dynamical systems lead one to construct and
define open and random systems: in both cases the question of (some sort of stability) is
one of the challenges. Demers’s article looks at the persistence of a spectral gap for the
transfer operator associated with the billiard map in the presence of small holes. The paper
by Shen and Van Strien studies stochastic stability, in the strong sense, for a class of maps of
the interval with a neutral fixed point and perturbed with additive noise. Finally the article
by Chen, Hu and Pesin addresses the question of the extent to which one kind of statistical
behaviour precludes another by showing natural examples of coexistence of hyperbolic and
non-hyperbolic behaviour in dynamics, in particular in smooth conservative systems.
Guest Editors
Anthony Quas (University of Victoria, Canada)
Sandro Vaienti (University of Toulon and Center of
Theoretical Physics, Luminy, Marseille)