We prove a central limit theorem for linear triangular arrays under weak dependence conditions. Our result is then applied to dependent random variables sampled by a $\mathbb{Z}$-valued transient random walk. This extends the results obtained by [N. Guillotin-Plantard and D. Schneider, Stoch. Dynamics3 (2003) 477–497]. An application to parametric estimation by random sampling is also provided.

Soit $f$ un difféomorphisme lisse de ${ℝ}_{+}$ fixant seulement l’origine, et ${𝒵}^r$ son centralisateur dans le groupe des difféomorphismes ${𝒞}^r$. Des résultat classiques de Kopell et Szekeres montrent que ${𝒵}^1$ est toujours un groupe à un paramètre. En revanche, Sergeraert a construit un $f$ dont le centralisateur ${𝒵}^{\infty }$ est réduit au groupe des itérés de $f$. On présente ici le résultat principal de [3] : ${𝒵}^{\infty }$ peut en fait être un sous-groupe propre et non-dénombrable (donc dense) de ${𝒵}^1$.

In this paper we extend results of Blokh, Bruckner, Humke and Sm’ıtal [Trans. Amer. Math. Soc. 348 (1996), 1357–1372] about characterization of $\omega$-limit sets from the class $𝒞(I,I)$ of continuous maps of the interval to the class $𝒞(𝕊,𝕊)$ of continuous maps of the circle. Among others we give geometric characterization of $\omega$-limit sets and then we prove that the family of $\omega$-limit sets is closed with respect to the Hausdorff metric.

La notion de type géométrique d’une partition de Markov est au centre de la classification des difféomorphismes de Smale i.e. des difféomorphismes $C^1$- structurellement stables des surfaces. On résout ici le problème de réalisabilité : on donne un critère effectif pour décider si une combinatoire abstraite est, ou n’est pas, le type géométrique d’une partition de Markov de pièce basique de difféomorphisme de Smale de surface compacte.

Nous étudions une classe de suites symboliques, les codages de rotations, intervenant dans des problèmes de répartition des suites ${\left(n\alpha \right)}_{n\in \mathbb{N}}$ et représentant une généralisation géométrique des suites sturmiennes. Nous montrons que ces suites peuvent être obtenues par itération de quatre substitutions définies sur un alphabet à trois lettres, puis en appliquant un morphisme de projection. L’ordre d’itération de ces applications est gouverné par un développement bi-dimensionnel de type “fraction continue”...

We give a characterization of the geometric automorphisms in a certain class of (not necessarily irreducible) free group automorphisms. When the automorphism is geometric, then it is induced by a pseudo-Anosov homeomorphism without interior singularities. An outer free group automorphism is given by a $1$-cocycle of a $2$-complex (a standard dynamical branched surface, see [7] and [9]) the fundamental group of which is the mapping-torus group of the automorphism. A combinatorial construction elucidates...

We study the combinatorics of distance doubling maps on the circle ℝ/ℤ with prototypes h(β) = 2β mod 1 and h̅(β) = -2β mod 1, representing the orientation preserving and orientation reversing case, respectively. In particular, we identify parts of the circle where the iterates $f^{\circ n}$ of a distance doubling map f exhibit “distance doubling behavior”. The results include well known statements for h related to the structure of the Mandelbrot set M. For h̅ they suggest some analogies to the structure of...

In this paper we study the commutativity property for topological sequence entropy. We prove that if $X$ is a compact metric space and $f,g:X\to X$ are continuous maps then $h_A(f\circ g)=h_A(g\circ f)$ for every increasing sequence $A$ if $X=[0,1]$, and construct a counterexample for the general case. In the interim, we also show that the equality $h_A\left(f\right)=h_A(f{|}_{{\cap }_{n\ge 0}{f}^n\left(X\right)})$ is true if $X=[0,1]$ but does not necessarily hold if $X$ is an arbitrary compact metric space.

Let K be a compact connected subset of cc, not reduced to a point, and F a univalent map in a neighborhood of K such that F(K) = K. This work presents a study and a classification of the dynamics of F in a neighborhood of K. When ℂ K has one or two connected components, it is proved that there is a natural rotation number associated with the dynamics. If this rotation number is irrational, the situation is close to that of “degenerate Siegel disks” or “degenerate Herman rings” studied by R. Pérez-Marco...

We combine some results from the literature to give examples of completely mixing interval maps without limit measure.

Let $W\left(I\right)$ denote the family of continuous maps $f$ from an interval $I=[a,b]$ into itself such that (1) $f\left(a\right)=f\left(b\right)\in \{a,b\}$; (2) they consist of two monotone pieces; and (3) they have periodic points of periods exactly all powers of $2$. The main aim of this paper is to compute explicitly the topological sequence entropy $h_D\left(f\right)$ of any map $f\in W\left(I\right)$ respect to the sequence $D={\left(2^{m-1}\right)}_{m=1}^{\infty }$.