We consider measure-preserving diffeomorphisms of the torus with zero entropy. We prove that every ergodic $C^1$-diffeomorphism with linear growth of the derivative is algebraically conjugate to a skew product of an irrational rotation on the circle and a circle $C^1$-cocycle. We also show that for no positive β ≠ 1 does there exist an ergodic $C^2$-diffeomorphism whose derivative has polynomial growth with degree β.

A new class of dynamical systems is defined, the class of “locally equicontinuous systems” (LE). We show that the property LE is inherited by factors as well as subsystems, and is closed under the operations of pointed products and inverse limits. In other words, the locally equicontinuous functions in $l_{\infty }\left(\mathbb{Z}\right)$ form a uniformly closed translation invariant subalgebra. We show that WAP ⊂ LE ⊂ AE, where WAP is the class of weakly almost periodic systems and AE the class of almost equicontinuous systems....

In this paper, we are interested in a diffusion process based on a gradient descent. The process is non Markov and has a memory term which is built as a weighted average of the drift term all along the past of the trajectory. For this type of diffusion, we study the long time behaviour of the process in terms of the memory. We exhibit some conditions for the long-time stability of the dynamical system and then provide, when stable, some convergence properties of the occupation measures and of the...

Viene considerata una classe di sistemi dinamici del toro bidimensionale $T^2$ . Tali sistemi presentano la forma di un prodotto skew fra l'endomorfismo Bernoulli $B_p\left(x\right)=\mathrm{mod}px,1$, $p\in \mathbb{Z}\setminus \left\{-1,0,1\right\}$, definito sul toro undidimensionale $T^1\equiv \left[0,1\right)$ ed una traslazione del toro stesso. Si dimostra che gli esponenti di Liapunov e l'entropia di Kolmogorov-Sinai della misura di Haar invariante possono essere calcolati esplicitamente. Viene infine discusso il decadimento delle correlazioni per i caratteri.

The paper deals with the notion of entropy for doubly stochastic operators. It is shown that the entropy defined by Maličky and Riečan in [MR] is equal to the operator entropy proposed in [DF]. Moreover, some continuity properties of the [MR] entropy are established.

The invariant measures for a Markovian operator corresponding to a random walk, in a random stationary one-dimensional environment defined by a dynamical system, are quasi-invariant measures for the system. We discuss the construction of such measures in the general case and show unicity, under some assumptions, for a rotation on the circle.

Fibre expanding systems have been introduced by Denker and Gordin. Here we show the existence of a finite partition for such systems which is fibrewise a Markov partition. Such partitions have direct applications to the Abramov-Rokhlin formula for relative entropy and certain polynomial endomorphisms of ℂ².

Soient $\Gamma$ un groupe discret géométriquement fini d’isométries d’une variété de Hadamard pincée $X$ et $𝒫$ une pointe de l’orbifold associé $ℳ:=\Gamma \setminus X$. Munissant $T^1ℳ$ de sa mesure de Patterson-Sullivan $m$, nous obtenons une estimation asymptotique de la masse d’un petit voisinage horocyclique de $𝒫$, moyennant une hypothèse sur la croissance du sous-groupe parabolique associé à $𝒫$, hypothèse qui est réalisée si $X$ est symétrique de rang $1$. Nous en déduisons une estimation asymptotique du temps de retour du flot géodésique...

Let $G:=\mathrm{SO}{(n,1)}^{\circ }$ and $\Gamma (n-1)/2$ for $n=2,3$ and when $\delta >n-2$ for $n\ge 4$, we obtain an effective archimedean counting result for a discrete orbit of $\Gamma$ in a homogeneous space $H\setminus G$ where $H$ is the trivial group, a symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family $\{{ℬ}_T\subset H\setminus G\}$ of compact subsets, there exists $\eta >0$ such that $\#\left[e\right]\Gamma \cap {ℬ}_T=ℳ\left({ℬ}_T\right)+O\left(ℳ{\left({ℬ}_T\right)}^{1-\eta }\right)$ for an explicit measure $ℳ$ on $H\setminus G$ which depends on $\Gamma$. We also apply the affine sieve and describe the distribution of almost primes on orbits of $\Gamma$ in arithmetic settings....

We prove that an invertible zero-dimensional dynamical system has an invariant measure of maximal entropy if and only if it is an extension of an asymptotically h-expansive system of equal topological entropy.

Let T be a Markov operator on an L¹-space. We study conditions under which T is mean ergodic and satisfies dim Fix(T) < ∞. Among other things we prove that the sequence $\left(n^{-1}{\sum }_{k=0}^{n-1}{T}^k\right)ₙ$ converges strongly to a rank-one projection if and only if there exists a function 0 ≠ h ∈ L¹₊ which satisfies $li{m}_{n\to \infty }\left|\right|(h-n^{-1}{\sum }_{k=0}^{n-1}{T}^{k}f)₊\left|\right|=0$ for every density f. Analogous results for strongly continuous semigroups are given.