Mean square value of exponential sums related to the representation of integers as sums of squares
Metric with ergodic geodesic flow is completely determined by unparameterized geodesics.
Nombre de rotation, mesures invariantes et ratio set des homéomorphismes affines par morceaux du cercle
Etant donné irrationnel de type constant, nous donnons des conditions explicites et génériques sur les pentes d’un homéomorphisme affine par morceaux du cercle de nombre de rotation , qui garantissent que la mesure de probabilité -invariante est singulière par rapport à la mesure de Haar. Cet article contient une preuve élémentaire d’un résultat de E. Ghys et V. Sergiescu : ”le nombre de rotation d’un homéomorphisme dyadique est rationnel”. Nous y étudions aussi le ratio set des homéomorphismes...
On diffeomorphisms with polynomial growth of the derivative on surfaces
We consider zero entropy -diffeomorphisms on compact connected -manifolds. We introduce the notion of polynomial growth of the derivative for such diffeomorphisms, and study it for diffeomorphisms which additionally preserve a smooth measure. We show that if a manifold M admits an ergodic diffeomorphism with polynomial growth of the derivative then there exists a smooth flow with no fixed point on M. Moreover, if dim M = 2, then necessarily M = ² and the diffeomorphism is -conjugate to a skew...
On disjointness properties of some smooth flows
Special flows over some locally rigid automorphisms and under L² ceiling functions satisfying a local L² Denjoy-Koksma type inequality are considered. Such flows are proved to be disjoint (in the sense of Furstenberg) from mixing flows and (under some stronger assumption) from weakly mixing flows for which the weak closure of the set of all instances consists of indecomposable Markov operators. As applications we prove that ∙ special flows built over ergodic interval exchange...
On natural invariant measures on generalised iterated function systems.
On symmetric logarithm and some old examples in smooth ergodic theory
We give a positive answer to the problem of existence of smooth weakly mixing but not mixing flows on some surfaces. More precisely, on each compact connected surface whose Euler characteristic is even and negative we construct smooth weakly mixing flows which are disjoint in the sense of Furstenberg from all mixing flows and from all Gaussian flows.
Orbit counting for some discrete groups acting on simply connected manifolds with negative curvature.
Parapuzzle of the multibrot set and typical dynamics of unimodal maps
We study the parameter space of unicritical polynomials . For complex parameters, we prove that for Lebesgue almost every , the map is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every , the map is either hyperbolic, or Collet–Eckmann, or infinitely renormalizable. These results are based on controlling the spacing between consecutive elements in the “principal nest” of parapuzzle pieces.
Physical measures for infinite-modal maps
We analyze certain parametrized families of one-dimensional maps with infinitely many critical points from the measure-theoretical point of view. We prove that such families have absolutely continuous invariant probability measures for a positive Lebesgue measure subset of parameters. Moreover, we show that both the density of such a measure and its entropy vary continuously with the parameter. In addition, we obtain exponential rate of mixing for these measures and also show that they satisfy the...
Propriétés ergodiques du flot horocyclique d'une surface hyperbolique géométriquement finie
Random perturbations and statistical properties of Hénon-like maps
Rigidity of smooth one-sided Bernoulli endomorphisms.
Semidefinite characterisation of invariant measures for one-dimensional discrete dynamical systems
Using recent results on measure theory and algebraic geometry, we show how semidefinite programming can be used to construct invariant measures of one-dimensional discrete dynamical systems (iterated maps on a real interval). In particular we show that both discrete measures (corresponding to finite cycles) and continuous measures (corresponding to chaotic behavior) can be recovered using standard software.
Sinai-Ruelle-Bowen measures for contracting Lorenz maps and flows
Sinai-Ruelle-Bowen measures for piecewise hyperbolic transformations.
Smoothness of invariant density for expanding transformations in higher dimensions.
Some continuity and approximation properties of a countable iterated function system.
SRB measures for non-hyperbolic systems with multidimensional expansion
Statistical properties of topological Collet–Eckmann maps