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 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.

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 $D={\left(2^{m-1}\right)}_{m=1}^{\infty }$.

We demonstrate that the set of topologically distinct inverse limit spaces of tent maps with a Cantor set for its postcritical ω-limit set has cardinality of the continuum. The set of folding points (i.e. points at which the space is not homeomorphic to the product of a zero-dimensional set and an arc) of each of these spaces is also a Cantor set.