We show that the set of conical points of a rational function of the Riemann sphere supports at most one conformal measure. We then study the problem of existence of such measures and their ergodic properties by constructing Markov partitions on increasing subsets of sets of conical points and by applying ideas of the thermodynamic formalism.

We study certain symmetries that arise when automorphisms S and T defined on a Lebesgue probability space (X, ℱ, μ) satisfy the equation $ST=T^{-1}S$. In an earlier paper [6] it was shown that this puts certain constraints on the spectrum of T. Here we show that it also forces constraints on the spectrum of $S^2$. In particular, $S^2$ has to have a multiplicity function which only takes even values on the orthogonal complement of the subspace $f\in L^2(X,ℱ,\mu ):f\left(T^{2}x\right)=f\left(x\right)$. For S and T ergodic satisfying this equation further constraints arise,...

Rauzy fractals are compact sets with fractal boundary that can be associated with any unimodular Pisot irreducible substitution. These fractals can be defined as the Hausdorff limit of a sequence of compact sets, where each set is a renormalized projection of a finite union of faces of unit cubes. We exploit this combinatorial definition to prove the connectedness of the Rauzy fractal associated with any finite product of three-letter Arnoux–Rauzy substitutions.

We obtain conditions for L₂ and strong consistency of the least square estimators of the coefficients in a multi-linear regression model with a stationary random noise. For given non-random regressors, we obtain conditions which ensure L₂-consistency for all wide sense stationary noise sequences with spectral measure in a given class. The condition for the class of all noises with continuous (i.e., atomless) spectral measures yields also $L_p$-consistency when the noise is strict sense stationary with...

We show that if $\Gamma$ is a discrete subgroup of the group of the isometries of ${ℍ}^k$, and if $\rho$ is a representation of $\Gamma$ into the group of the isometries of ${ℍ}^n$, then any $\rho$-equivariant map $F:{ℍ}^k\to {ℍ}^n$ extends to the boundary in a weak sense in the setting of Borel measures. As a consequence of this fact, we obtain an extension of a result of Besson, Courtois and Gallot about the existence of volume non-increasing, equivariant maps. Then, we show that the weak extension we obtain is actually a measurable $\rho$-equivariant...

We elaborate a method allowing the determination of 0-1 matrices corresponding to dynamics of the interval having stable, 2k-periodic orbits, k belonging to N. By recurrence on the finite dimensional matrices, we establish the form of the infinite matrices (k --> ∞).

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...

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...

$M_1$ 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 $u\mapsto \int {\left|\text{grad}u\right|}^2$ 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...

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 ${\sum }_{k\ge 0}{k}^r{P}^{k}f$, $r\in \mathbb{N}$, under some regularity assumptions and implies the central limit theorem with a rate in $n^{-\frac{1}{2}}$ for the corresponding Markov chain. An application to a non uniformly hyperbolic transformation on the interval is also given.

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, $r\in \mathbb{N}$, under some regularity assumptions and implies the central limit theorem with a rate in $n^{-\frac{1}{2}}$ for the corresponding Markov chain. An application to a non uniformly hyperbolic transformation on the interval is also given.

We prove the norm convergence of multiple ergodic averages along cubes for several commuting transformations, and derive corresponding combinatorial results. The method we use relies primarily on the "magic extension" established recently by B. Host.

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).