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Construction of non-constant and ergodic cocycles

Mahesh Nerurkar (2000)

Colloquium Mathematicae

We construct continuous G-valued cocycles that are not cohomologous to any compact constant via a measurable transfer function, provided the underlying dynamical system is rigid and the range group G satisfies a certain general condition. For more general ergodic aperiodic systems, we also show that the set of continuous ergodic cocycles is residual in the class of all continuous cocycles provided the range group G is a compact connected Lie group. The first construction is based on the "closure...

Continuous subadditive processes and formulae for Lyapunov characteristic exponents

Wojciech Słomczyński (1995)

Annales Polonici Mathematici

Asymptotic properties of various semidynamical systems can be examined by means of continuous subadditive processes. To investigate such processes we consider different types of exponents: characteristic, central, singular and global exponents and we study their properties. We derive formulae for central and singular exponents and show that they provide upper bounds for characteristic exponents. The concept of conjugate processes introduced in this paper allows us to find lower bounds for characteristic...

Convergence and uniqueness problems for Dirichlet forms on fractals

Roberto Peirone (2000)

Bollettino dell'Unione Matematica Italiana

M 1 è un particolare operatore di minimizzazione per forme di Dirichlet definite su un sottoinsieme finito di un frattale K che è, in un certo senso, una sorta di frontiera di K . Viene talvolta chiamato mappa di rinormalizzazione ed è stato usato per definire su K un analogo del funzionale u ↦ ∫ grad ⁢ u 2 e un moto Browniano. In questo lavoro si provano alcuni risultati sull'unicità dell'autoforma (rispetto a M 1 ), e sulla convergenza dell'iterata di M 1 rinormalizzata. Questi risultati sono collegati con l'unicità...

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.

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 ∑k≥0krPkƒ, 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.

Correlation asymptotics from large deviations in dynamical systems with infinite measure

Sébastien Gouëzel (2011)

Colloquium Mathematicae

We extend a result of Doney [Probab. Theory Related Fields 107 (1997)] on renewal sequences with infinite mean to renewal sequences of operators. As a consequence, we get precise asymptotics for the transfer operator and for correlations in dynamical systems preserving an infinite measure (including intermittent maps with an arbitrarily neutral fixed point).

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