Let f:Tn→Tn (n ≥ 3) be a hyperbolic toral automorphism. Let A be the set of α > 0 such that there is a Hölder continuous path of index α in Tn with 1-dimensional orbit-closure under f We prove that α0 = sup A can be expressed in terms of the eigenvalues of f and that α0 ∈ A if and only if α0 < 1.

We prove that any locally compact, non-compact, second countable group acts minimally on any metrizable connected manifold modelled on the separable Hilbert space.

We prove a joint partial equidistribution result for common perpendiculars with given density on equidistributing equidistant hypersurfaces, towards a measure supported on truncated stable leaves. We recover a result of Marklof on the joint partial equidistribution of Farey fractions at a given density, and give several analogous arithmetic applications, including in Bruhat–Tits trees.

We introduce some notions of conditional mean dimension for a factor map between two topological dynamical systems and discuss their properties. With the help of these notions, we obtain an inequality to estimate the mean dimension of an extension system. The conditional mean dimension for G-extensions is computed. We also exhibit some applications in dynamical embedding problems.

We introduce the study of the local dynamics around a parabolic indifferent invariant curve for fibered holomorphic maps. As in the classical non-fibered case, we show that petals are the main ingredient. Nevertheless, one expects that the properties of the base rotation number should play an important role in the arrangement of the petals. We exhibit examples where the existence and the number of petals depend not just on the complex coordinate of the map, but on the base rotation number. Furthermore, under additional hypotheses on the arithmetic and smoothness of the map, we present a theorem that allows a characterization of the local dynamics around a parabolic invariant curve.

Let denote the class of transformations with the weak Pinsker property, and let []m denote the class of transformations m-equivalent to some member of , where m is an entropy-preserving size. We show that if T is a factor of an element of []m, then T ∈ []m, and if T is an m-limit of elements of []m, then T ∈ []m.

Let (X, F, μ) be a probability measure space, p and β real numbers such that 1≤p<+∞ and 0<β<p. For any linear positive operator T satisfying T1, T*1 = 1 we prove the norm and pointwise convergence of the sequence We get then the pointwise and norm convergence in Lp, 0 < β ≥ 1 < p < 2, of the sequence sgn Sif for any positive linear operator on Lp(Ω, A, μ) (μ-σ-finite) verifying ∥(1 − α)I + αS∥p ≤ 1 for a real number 0 < α < 1. In the particular case α = 1, (S is a contraction), β = p−l, this result gives the pointwise and norm convergence of the sequences introduced by Beauzamy and Enflo in 1985 to the asymptotic center of the sequence .

Bishop [2]has given a proof of Birkhoff's ergodic theorem by establishing upcrossing inequalities similar to those of Doob. Such inequalities can be considered as quantitative improvements of convergence theorems: while convergence a.e. means that the number of upcrossings of any interval is a.e. finite, they assert integrability and prove bounds for the integrals. The main point of this paper is to prove upcrossing inequalities for the class of subadditive superstationary processes introduced by Abid [1] as a common generalization of Kingman's [5] subadditive stationary processes and Krengel's [6] superstationary processes. We make use of ideas of Smeltzer [7] who handled the subadditive stationary discrete parameter case in his unpublished thesis. In the continuous parameter case our upcrossing inequality requires more restrictive conditions than the corresponding convergence theorem, due to Hachem [3]. We actually show by example that the number of upcrossings need not be integrable under the assumptions of Hachem even for additive stationary processes.