Random dynamics and its applications.
Random entropy and recurrence.
Random orderings and unique ergodicity of automorphism groups
We show that the only random orderings of finite graphs that are invariant under isomorphism and induced subgraph are the uniform random orderings. We show how this implies the unique ergodicity of the automorphism group of the random graph. We give similar theorems for other structures, including, for example, metric spaces. These give the first examples of uniquely ergodic groups, other than compact groups and extremely amenable groups, after Glasner andWeiss’s example of the group of all permutations...
Random permutations and unique fully supported ergodicity for the Euler adic transformation
There is only one fully supported ergodic invariant probability measure for the adic transformation on the space of infinite paths in the graph that underlies the eulerian numbers. This result may partially justify a frequent assumption about the equidistribution of random permutations.
Random perturbations and statistical properties of Hénon-like maps
Rank-one group actions with simple mixing -subactions.
Rate of convergence of conditional entropies for some maps of an interval
Ratner's property for special flows over irrational rotations under functions of bounded variation. II
We consider special flows over the rotation on the circle by an irrational α under roof functions of bounded variation. The roof functions, in the Lebesgue decomposition, are assumed to have a continuous singular part coming from a quasi-similar Cantor set (including the devil's staircase case). Moreover, a finite number of discontinuities is allowed. Assuming that α has bounded partial quotients, we prove that all such flows are weakly mixing and enjoy the weak Ratner property. Moreover, we provide...
Reducibility and point spectrum for linear quasi-periodic skew-products.
Réductibilité presque partout des flots fibrés quasi-périodiques à valeurs dans des groupes compacts
Refinement type equations: sources and results
It has been proved recently that the two-direction refinement equation of the form can be used in wavelet theory for constructing two-direction wavelets, biorthogonal wavelets, wavelet packages, wavelet frames and others. The two-direction refinement equation generalizes the classical refinement equation , which has been used in many areas of mathematics with important applications. The following continuous extension of the classical refinement equation has also various interesting applications....
Regularity of local manifolds in dispersing billiards.
Regularity of measure theoretic entropy for geodesic flows of negative curvature: I.
Regularity of topological and metric entropy of hyperbolic flows.
Relating Hausdorff measures and harmonic measures on parabolic Jordan curves.
Relative property (T) and linear groups
Relative property (T) has recently been used to show the existence of a variety of new rigidity phenomena, for example in von Neumann algebras and the study of orbit-equivalence relations. However, until recently there were few examples of group pairs with relative property (T) available through the literature. This motivated the following result: A finitely generated group admits a special linear representation with non-amenable -Zariski closure if and only if it acts on an Abelian group (of...
Relative spectral theory and measure-theoretic entropy of gaussian extensions
We describe the natural framework in which the relative spectral theory is developed. We give some results and indicate how they relate to two open problems in ergodic theory. We also compute the relative entropy of gaussian extensions of ergodic transformations.
Remarks on the tightness of cocycles
We prove a generalised tightness theorem for cocycles over an ergodic probability preserving transformation with values in Polish topological groups. We also show that subsequence tightness of cocycles over a mixing probability preserving transformation implies tightness. An example shows that this latter result may fail for cocycles over a mildly mixing probability preserving transformation.
Residuality of dynamical morphisms
We present a unified approach to the finite generator theorem of Krieger, the homomorphism theorem of Sinai and the isomorphism theorem of Ornstein. We show that in a suitable space of measures those measures which define isomorphisms or respectively homomorphisms form residual subsets.
Return time statistics for unimodal maps
We prove that a non-flat S-unimodal map satisfying a weak summability condition has exponential return time statistics on intervals around a.e. point. Moreover we prove that the convergence to the entropy in the Ornstein-Weiss formula enjoys normal fluctuations.