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Convergence of iterates of a transfer operator, application to dynamical systems and to Markov chains

Jean-Pierre Conze;  Albert Raugi — 2003

ESAIM: Probability and Statistics

We present a spectral theory for a class of operators satisfying a weak “Doeblin–Fortet” condition and apply it to a class of transition operators. This gives the convergence of the series ∑ k ≥ 0 k r P k f , r ∈ ℕ , under some regularity assumptions and implies the central limit theorem with a rate in n - 1 2 for the corresponding Markov chain. An application to a non uniformly hyperbolic transformation on the interval is also given.

On the ergodic decomposition for a cocycle

Jean-Pierre Conze;  Albert Raugi — 2009

Colloquium Mathematicae

Let (X,,μ,τ) be an ergodic dynamical system and φ be a measurable map from X to a locally compact second countable group G with left Haar measure m G . We consider the map τ φ defined on X × G by τ φ : ( x , g ) ↦ ( τ x , φ ( x ) g ) and the cocycle ( φ ₙ ) n ∈ ℤ generated by φ. Using a characterization of the ergodic invariant measures for τ φ , we give the form of the ergodic decomposition of μ ( d x ) ⊗ m G ( d g ) or more generally of the τ φ -invariant measures μ χ ( d x ) ⊗ χ ( g ) m G ( d g ) , where μ χ ( d x ) is χ∘φ-conformal for an exponential χ on G.

Almost everywhere convergence of convolution powers on compact abelian groups

Jean-Pierre Conze;  Michael Lin — 2013

Annales de l'I.H.P. Probabilités et statistiques

It is well-known that a probability measure μ on the circle 𝕋 satisfies ∥ μ n * f - ∫ f d m ∥ p → 0 for every f ∈ L p , every (some) p ∈ [ 1 , ∞ ) , if and only if | μ ^ ( n ) | l t ; 1 for every non-zero n ∈ ℤ ( μ is strictly aperiodic). In this paper we study the a.e. convergence of μ n * f for every f ∈ L p whenever p g t ; 1 . We prove a necessary and sufficient condition, in terms of the Fourier–Stieltjes coefficients of μ , for the strong sweeping out property (existence of a Borel set B with lim sup μ n * 1 B = 1 a.e. and lim inf μ n * 1 B = 0 a.e.). The results are extended to general compact Abelian groups G with Haar...

Régularité des bases d'ondelettes et mesures ergodiques.

Albert Cohen;  Jean-Pierre Conze — 1992

Revista Matemática Iberoamericana

Nous reprenons la construction des bases orthonormées d'ondelettes à partir des filtres miroirs en quadrature tel qu'elle apparaît dans [4]. Nous montrons que leur régularité est liée à une mesure invariante pour la transformation ω → 2ω mod-2π. Cette méthode permet d'obtenir le facteur exact qui relie asymptotiquement la régularité des ondelettes constriutes dans [4] à la taille de leur support.

Convergence of iterates of a transfer operator, application to dynamical systems and to Markov chains

Jean-Pierre Conze;  Albert Raugi — 2010

ESAIM: Probability and Statistics

We present a spectral theory for a class of operators satisfying a weak “Doeblin–Fortet" condition and apply it to a class of transition operators. This gives the convergence of the series ∑, r ∈ ℕ , under some regularity assumptions and implies the central limit theorem with a rate in n - 1 2 for the corresponding Markov chain. An application to a non uniformly hyperbolic transformation on the interval is also given.

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