Sur les retardateurs
Symbolic discrepancy and self-similar dynamics
We consider subshifts arising from primitive substitutions, which are known to be uniquely ergodic dynamical systems. In order to precise this point, we introduce a symbolic notion of discrepancy. We show how the distribution of such a subshift is in part ruled by the spectrum of the incidence matrices associated with the underlying substitution. We also give some applications of these results in connection with the spectral study of substitutive dynamical systems.
The asymptotics of the Perron-Frobenius operator of a class of interval maps preserving infinite measures
We determine the asymptotic behaviour of the iterates of the Perron-Frobenius operator for specific interval maps with an indifferent fixed point which gives rise to an infinite invariant measure.
The Billiard in a Regular Polygon.
The branch locus for one-dimensional Pisot tiling spaces
If φ is a Pisot substitution of degree d, then the inflation and substitution homeomorphism Φ on the tiling space factors via geometric realization onto a d-dimensional solenoid. Under this realization, the collection of Φ-periodic asymptotic tilings corresponds to a finite set that projects onto the branch locus in a d-torus. We prove that if two such tiling spaces are homeomorphic, then the resulting branch loci are the same up to the action of certain affine maps on the torus.
The classification of tiling space flows.
The rate of convergence for iterated function systems
Iterated function systems with place-dependent probabilities are considered. It is shown that the rate of convergence of transition probabilities to a unique invariant measure is geometric.
The uniqueness of invariant measures for Markov operators
It is shown that Markov operators with equicontinuous dual operators which overlap supports have at most one invariant measure. In this way we extend the well known result proved for Markov operators with the strong Feller property by R. Z. Khas'minski.
The vector individual weighted ergodic theorem for bounded Besicovich sequences.
Tower multiplexing and slow weak mixing
A technique is presented for multiplexing two ergodic measure preserving transformations together to derive a third limiting transformation. This technique is used to settle a question regarding rigidity sequences of weak mixing transformations. Namely, given any rigidity sequence for an ergodic measure preserving transformation, there exists a weak mixing transformation which is rigid along the same sequence. This establishes a wide range of rigidity sequences for weakly mixing dynamical systems....
Truncated Gamow functions and the exponential decay law
Une condition asymptotique pour le calcul de constantes de Sobolev logarithmiques sur la droite
On présente une formule explicite pour la constante de Sobolev logarithmique correspondant à des diffusions réelles ou à des processus entiers de vie et de mort, sous l’hypothèse que certaines quantités, naturellement associées à des inégalités de Hardy dans ce contexte, approchent leur supremum au bord de leur domaine de définition. La preuve se ramène au cas de la constante de Poincaré, à l’aide de comparaisons exactes entre entropie et variances appropriées.
Uniform exponential ergodicity of stochastic dissipative systems
We study ergodic properties of stochastic dissipative systems with additive noise. We show that the system is uniformly exponentially ergodic provided the growth of nonlinearity at infinity is faster than linear. The abstract result is applied to the stochastic reaction diffusion equation in with .
Unique Bernoulli -measures
We improve and subsume the conditions of Johansson and Öberg and Berbee for uniqueness of a -measure, i.e., a stationary distribution for chains with complete connections. In addition, we prove that these unique -measures have Bernoulli natural extensions. We also conclude that we have convergence in the Wasserstein metric of the iterates of the adjoint transfer operator to the -measure.
Vitesse dans le théorème limite central pour certains systèmes dynamiques quasi-hyperboliques
Nous présentons une méthode permettant d’établir le théorème limite central avec vitesse en pour certains systèmes dynamiques. Elle est basée sur une propriété de décorrélation forte qui semble assez naturelle dans le cadre des systèmes quasi-hyperboliques. Nous prouvons que cette propriété est satisfaite par les exemples des flots diagonaux sur un quotient compact de et les « transformations » non uniformément hyperboliques du tore étudiées par Shub et Wilkinson.
Vitesse de convergence dans le théorème limite central pour des chaînes de Markov fortement ergodiques
Soit Q une probabilité de transition sur un espace mesurable E, admettant une probabilité invariante, soit (Xn)n une chaîne de Markov associée à Q, et soit ξ une fonction réelle mesurable sur E, et Sn=∑nk=1ξ(Xk). Sous des hypothèses fonctionnelles sur l’action de Q et des noyaux de Fourier Q(t), nous étudions la vitesse de convergence dans le théorème limite central pour la suite . Selon les hypothèses nous obtenons une vitesse enn−τ/2 pour tout τ<1, ou bien en n−1/2. Nous appliquons la...
Weak almost periodicity of contractions and coboundaries of non-singular transformations
It is well known that a weakly almost periodic operator T in a Banach space is mean ergodic, and in the complex case, also λT is mean ergodic for every |λ|=1. We prove that a positive contraction on is weakly almost periodic if (and only if) it is mean ergodic. An example shows that without positivity the result is false. In order to construct a contraction T on a complex such that λT is mean ergodic whenever |λ|=1, but T is not weakly almost periodic, we prove the following: Let τ be an invertible...
Weak- estimates and ergodic theorems.
Weakly mixing rank-one transformations conjugate to their squares
Utilizing the cut-and-stack techniques we construct explicitly a weakly mixing rigid rank-one transformation T which is conjugate to T². Moreover, it is proved that for each odd q, there is such a T commuting with a transformation of order q. For any n, we show the existence of a weakly mixing T conjugate to T² and whose rank is finite and greater than n.
Алгебра симметрий и блочная диагонализуемость линейных квазипериодических систем блочно-треугольного вида