We study asymptotic properties of eigenfunctions of the Laplacian on compact Riemannian surfaces of Anosov type (for instance negatively curved surfaces). More precisely, we give an answer to a question of Anantharaman and Nonnenmacher [4] by proving that the Kolmogorov-Sinai entropy of a semiclassical measure $\mu$ for the geodesic flow $g^t$ is bounded from below by half of the Ruelle upper bound. (This text has been written for the proceedings of the ${37}^{\text{èmes}}$ Journées EDP (Port d’Albret-June, 7-11 2010))

A number of recent works have sought to generalize the Kolmogorov-Sinai entropy of probability-preserving transformations to the setting of Markov operators acting on the integrable functions on a probability space (X,μ). These works have culminated in a proof by Downarowicz and Frej that various competing definitions all coincide, and that the resulting quantity is uniquely characterized by certain abstract properties. On the other hand, Makarov has shown that this 'operator...

We consider scalar reaction-diffusion equations on bounded and extended domains, both with the autonomous and time-periodic nonlinear term. We discuss the meaning and implications of the ergodic Poincaré-Bendixson theorem to dynamics. In particular, we show that in the extended autonomous case, the space-time topological entropy is zero. Furthermore, we characterize in the extended nonautonomous case the space-time topological and metric entropies as entropies of a pair of commuting planar homeomorphisms....

Topological and metric entropy pairs of ℤ²-actions are defined and their properties are investigated, analogously to ℤ-actions. In particular, mixing properties are studied in connection with entropy pairs.

We study the notion of ε-independence of a process on finitely (or countably) many states and that of ε-independence between two processes defined on the same measure preserving transformation. For that we use the language of entropy. First we demonstrate that if a process is ε-independent then its ε-independence from another process can be verified using a simplified condition. The main direction of our study is to find natural examples of ε-independence. In case of ε-independence of one process,...

We give an introduction to adelic mixing and its applications for mathematicians knowing about the mixing of the geodesic flow on hyperbolic surfaces. We focus on the example of the Hecke trees in the modular surface.

Given a rational function $\varphi \left(T\right)$ on ${\mathbb{P}}^1$ of degree at least 2 with coefficients in a number field $k$, we show that for each place $v$ of $k$, there is a unique probability measure ${\mu }_{\varphi ,v}$ on the Berkovich space ${\mathbb{P}}_{\mathrm{Berk},\mathrm{v}}^1/{ℂ}_v$ such that if $\left\{z_n\right\}$ is a sequence of points in ${\mathbb{P}}^1\left(\overline{k}\right)$ whose $\varphi$-canonical heights tend to zero, then the $z_n$’s and their $\mathrm{Gal}(\overline{k}/k)$-conjugates are equidistributed with respect to ${\mu }_{\varphi ,v}$.The proof uses a polynomial lift $F(x,y)=(F_1(x,y),F_2(x,y))$ of $\varphi$ to construct a two-variable Arakelov-Green’s function $g_{\varphi ,v}(x,y)$ for each $v$. The measure ${\mu }_{\varphi ,v}$ is obtained by taking the...

Basic ergodic properties of the ELF class of automorphisms, i.e. of the class of ergodic automorphisms whose weak closure of measures supported on the graphs of iterates of T consists of ergodic self-joinings are investigated. Disjointness of the ELF class with: 2-fold simple automorphisms, interval exchange transformations given by a special type permutations and time-one maps of measurable flows is discussed. All ergodic Poisson suspension automorphisms as well as dynamical systems determined...

If the ergodic transformations S, T generate a free ${\mathbb{Z}}^2$ action on a finite non-atomic measure space (X,S,µ) then for any $c_1,c_2\in ℝ$ there exists a measurable function f on X for which ${\left(N+1\right)}^{-1}{\sum }_{j=0}^{N}f\left(S^{j}x\right)\to c_1$ and ${(N+1)}^{-1}{\sum }_{j=0}^{N}f\left(T^{j}x\right)\to c_2\mu$-almost everywhere as N → ∞. In the special case when S, T are rationally independent rotations of the circle this result answers a question of M. Laczkovich.

We study the convergence of the ergodic averages $\frac{1}{N}{\sum }_{k=0}^{N-1}\theta \left(k\right)f\circ T^{u_k}$ where ${\left(\theta \left(k\right)\right)}_{k\in \mathbb{N}}$ is a bounded sequence and ${\left(u_k\right)}_{k\in \mathbb{N}}$ a strictly increasing sequence of integers such that ${\mathrm{Sup}}_{\alpha \in ℝ}\left|{\sum }_{k=0}^{N-1}\theta \left(k\right)\exp \left(2i\pi \alpha u_k\right)\right|=O\left(N^{\delta }\right)$ for some $\delta \<1$. Moreover we give explicit such sequences $\theta$ and $u$ and we investigate in particular the case where $\theta$ is a $q$-multiplicative sequence.

Two types of weighted ergodic averages are studied. It is shown that if F = {Fₙ} is an admissible superadditive process relative to a measure preserving transformation, then a Wiener-Wintner type result holds for F. Using this result new good classes of weights generated by such processes are obtained. We also introduce another class of weights via the group of unitary functions, and study the convergence of the corresponding weighted averages. The limits of such weighted averages are also identified....

The purpose of this note is to prove various versions of the ergodic decomposition theorem for probability measures on standard Borel spaces which are quasi-invariant under a Borel action of a locally compact second countable group or a discrete nonsingular equivalence relation. In the process we obtain a simultaneous ergodic decomposition of all quasi-invariant probability measures with a prescribed Radon-Nikodym derivative, analogous to classical results about decomposition of invariant probability...

We define a class of discrete Abelian group extensions of rank-one transformations and establish necessary and sufficient conditions for these extensions to be power weakly mixing. We show that all members of this class are multiply recurrent. We then study conditions sufficient for showing that Cartesian products of transformations are conservative for a class of invertible infinite measure-preserving transformations and provide examples of these transformations.

We construct a natural invariant measure concentrated on the set of square-free numbers, and invariant under the shift. We prove that the corresponding dynamical system is isomorphic to a translation on a compact, Abelian group. This implies that this system is not weakly mixing and has zero measure-theoretical entropy.