Soit $f$ un homéomorphisme du plan qui préserve l’orientation et qui a un point périodique $z^{*}$ de période $q\ge 2$. Nous montrons qu’il existe un point fixe $z$ tel que le nombre d’enlacement de $z^{*}$ et $z$ ne soit pas nul. En d’autres termes, le nombre de rotation de l’orbite de $z^{*}$ dans l’anneau ${ℝ}^2\setminus \left\{z\right\}$ est un élément non nul de $ℝ/\mathbb{Z}$. Ceci donne une réponse positive à une question posée par John Franks.

We deal with a subshift of finite type and an equilibrium state μ for a Hölder continuous function. Let αⁿ be the partition into cylinders of length n. We compute (in particular we show the existence of the limit) $li{m}_{n\to \infty }{n}^{-1}log{\sum }_{j=0}^{\tau ₙ\left(x\right)}\mu \left(\alpha ⁿ\left(T^j\left(x\right)\right)\right)$, where $\alpha ⁿ\left(T^j\left(x\right)\right)$ is the element of the partition containing $T^j\left(x\right)$ and τₙ(x) is the return time of the trajectory of x to the cylinder αⁿ(x).

Let us consider the simplest model of one-dimensional probabilistic cellular automata (PCA). The cells are indexed by the integers, the alphabet is $\{0,1\}$, and all the cells evolve synchronously. The new content of a cell is randomly chosen, independently of the others, according to a distribution depending only on the content of the cell itself and of its right neighbor. There are necessary and sufficient conditions on the four parameters of such a PCA to have a Bernoulli product invariant measure....

In this paper we introduce a ⊗-operation over Markov transition matrices, in the context of subshift of finite type, reproducing symbolic properties of the iterates of the critical point on a one-parameter family of unimodal maps. To the *-product between kneading sequences we associate a ⊗-product between the corresponding Markov matrices.

In 1926 Birkhoff defined the center depth, one of the fundamental invariants that characterize the topological structure of a dynamical system. In this paper, we introduce the concepts of prolongational centers and their depths, which lead to a complete family of topological invariants. Some basic properties of the prolongational centers and their depths are established. Also, we construct a dynamical system in which the depth of a prolongational center is a prescribed countable ordinal.

This article investigates under what conditions nontransitivity can coexist with the asymptotic average shadowing property. We show that there is a large class of maps satisfying both conditions simultaneously and that it is possible to find such examples even among maps on a compact interval. We also study the limit shadowing property and its relation to the asymptotic average shadowing property.

Dans ce travail, nous construisons explicitement deux isomorphismes métriques partout continus. L’un entre le système dynamique symbolique associé à la substitution $\sigma :0\mapsto 01,1\mapsto 02,2\mapsto 0$ et une rotation sur le tore ${𝕋}^2$; l’autre, entre le système adique stationnaire [33] associé à la matrice de la substitution et la même rotation. Pour cela, nous étudions les propriétés arithmétiques de la frontière d’un ensemble compact de $ℂ$ appelé “fractal de Rauzy”. Les constructions se généralisent aux substitutions de la forme ${\sigma }_k:0\mapsto 01,1\mapsto 02,\cdots k-1\mapsto 0k,k\mapsto 0$...

Nous étudions certaines propriétés combinatoires, ergodiques et arithmétiques du point fixe de la substitution de Tribonacci (introduite par G. Rauzy) et de la rotation du tore ${𝕋}^2$ qui lui est associée. Nous établissons une généralisation géométrique du théorème des trois distances et donnons une formule explicite pour la fonction de récurrence du point fixe. Nous donnons des propriétés d’approximation diophantienne du vecteur de la rotation de ${𝕋}^2$ : nous montrons, que pour une norme adaptée, la suite...

A substitution φ is strong Pisot if its abelianization matrix is nonsingular and all eigenvalues except the Perron-Frobenius eigenvalue have modulus less than one. For strong Pisot φ that satisfies a no cycle condition and for which the translation flow on the tiling space ${}_{\phi }$ has pure discrete spectrum, we describe the collection ${}_{\phi }^P$ of pairs of proximal tilings in ${}_{\phi }$ in a natural way as a substitution tiling space. We show that if ψ is another such substitution, then ${}_{\phi }$ and ${}_{\psi }$ are homeomorphic if and...

Two dynamical deformation theories are presented – one for surface homeomorphisms, called pruning, and another for graph endomorphisms, called kneading – both giving conditions under which all of the dynamics in an open set can be destroyed, while leaving the dynamics unchanged elsewhere. The theories are related to each other and to Thurston’s classification of surface homeomorphisms up to isotopy.

Recall that a P-set is a closed set X such that the intersection of countably many neighborhoods of X is again a neighborhood of X. We show that if 𝔱 = 𝔠 then there is a minimal right ideal of (βℕ,+) that is also a P-set. We also show that the existence of such P-sets implies the existence of P-points; in particular, it is consistent with ZFC that no minimal right ideal is a P-set. As an application of these results, we prove that it is both consistent with and independent of ZFC that the shift...

A pullback incremental attraction, a nonautonomous version of incremental stability, is introduced for nonautonomous systems that may have unbounded limiting solutions. Its characterisation by a Lyapunov function is indicated.