We discuss main properties of the dynamics on minimal attraction centers (σ-limit sets) of single trajectories for continuous maps of a compact metric space into itself. We prove that each nowhere dense nonvoid closed set in ${ℝ}^n$, n ≥ 1, is a σ-limit set for some continuous map.

On appelle échange d’intervalles l’application qui consiste à réordonner les intervalles d’une partition de $[0,1[$ suivant une permutation donnée. Dans le cas des partitions en trois intervalles, nous donnons une caractérisation combinatoire des suites codant, d’après la partition définissant l’échange, l’orbite d’un point de $[0,1[$ sous l’action de cette transformation.

In this paper we address the following question due to Marcy Barge: For what f:I → I is it the case that the inverse limit of I with single bonding map f can be embedded in the plane so that the shift homeomorphism $\widehat{f}$ extends to a diffeomorphism ([BB, Problem 1.5], [BK, Problem 3])? This question could also be phrased as follows: Given a map f:I → I, find a diffeomorphism $F:{ℝ}^2\to {ℝ}^2$ so that F restricted to its full attracting set, ${\bigcap }_{k\ge 0}{F}^k\left({ℝ}^2\right)$, is topologically conjugate to $\widehat{f}:(I,f)\to (I,f)$. In this situation, we say that the inverse...

We show that an aperiodic minimal tiling space with only finitely many asymptotic composants embeds in a surface if and only if it is the suspension of a symbolic interval exchange transformation (possibly with reversals). We give two necessary conditions for an aperiodic primitive substitution tiling space to embed in a surface. In the case of substitutions on two symbols our classification is nearly complete. The results characterize the codimension one hyperbolic attractors of surface diffeomorphisms...

Un homéomorphisme de Brouwer est un homéomorphisme du plan, sans point fixe, préservant l’orientation. Le théorème des translations planes affirme qu’un tel homéomorphisme s’obtient toujours en « recollant des translations ». Dans cet article, nous introduisons un nouvel invariant de conjugaison des homéomorphismes de Brouwer, l’ensemble oscillant, pour tenter de décrire assez précisément la manière dont s’effectue le recollement des translations. D’une part, nous utilisons la notion d’ensemble...

Nous définissons la notion d’ensemble bien ordonné de torsion nulle pour les applications déviant la verticale. Contrairement aux études variationnelles de [14] et [1], nous proposons une approche topologique. On retrouve pour ces ensembles un grand nombre de propriétés des ensembles bien ordonnés décrites dans [11]. En reprenant un argument de G.Hall [7], nous montrons en particulier que pour tout nombre de rotation, il existe un ensemble bien ordonné de torsion nulle.

Let f be a continuous map of the circle $S^1$ or the interval I into itself, piecewise $C^1$, piecewise monotone with finitely many intervals of monotonicity and having positive entropy h. For any ε > 0 we prove the existence of at least $e^{(h-\epsilon )n_k}$ periodic points of period $n_k$ with large derivative along the period, $|{\left(f^{n_k}\right)}^{\text{'}}|>e^{(h-\epsilon )n_k}$ for some subsequence $n_k$ of natural numbers. For a strictly monotone map f without critical points we show the existence of at least $(1-\epsilon )e^{hn}$ such points.

Let $f:I\to I$ be a $C^2$ multimodal interval map satisfying polynomial growth of the derivatives along critical orbits. We prove the existence and uniqueness of equilibrium states for the potential ${\phi }_t:x\mapsto -tlog\left|Df\left(x\right)\right|$ for $t$ close to $1$, and also that the pressure function $t\mapsto P\left({\phi }_t\right)$ is analytic on an appropriate interval near $t=1$.

This work provides rates of convergence in the Darling-Kac law for infinite measure preserving Pomeau-Manneville (unit interval) maps. Along the way we obtain error rates for the stable law associated with the first return map and the first return time to some suitable set inside the unit interval.

Nous étudions un exemple de transformation non uniformément hyperbolique de l’intervalle $[0,1[$. Des exemples analogues ont été étudiés par de nombreux auteurs. Notre méthode utilise une théorie spectrale, pour une classe d’opérateurs vérifiant des conditions faibles de Doeblin-Fortet, introduite dans [1]. Elle nous permet, en particulier, de donner une estimation de la vitesse de décroissance des corrélations pour des fonctions non höldériennes.

We construct ${}^{\infty }$ maps T on the interval and on the circle which are Lebesgue exact preserving an absolutely continuous infinite measure μ ≪ λ, such that for any probability measure ν ≪ λ the sequence ${\left(n^{-1}{\sum }_{k=0}^{n-1}\nu \circ T^{-k}\right)}_{n\ge 1}$ of arithmetical averages of image measures does not converge weakly.

The example is constructed of the C1-smooth skew product of interval maps possessing the one-dimensional ramified continuum (containing no arcs homeomorphic to the circle) with an infinite set of ramification points as the global attractor.

We prove that for every ϵ > 0 there exists a minimal diffeomorphism f: ² → ² of class $C^{3-ϵ}$ and semiconjugate to an ergodic translation with the following properties: zero entropy, sensitivity to initial conditions, and Li-Yorke chaos. These examples are obtained through the holonomy of the unstable foliation of Mañé’s example of a derived-from-Anosov diffeomorphism on ³.

Le Calvez a montré que si $F$ est un homéomorphisme isotope à l’identité d’une surface $M$ admettant un relèvement $\tilde{F}$ au revêtement universel n’ayant pas de points fixes, alors il existe un feuilletage topologique de $M$ transverse à la dynamique. Nous montrons que ce résultat se généralise au cas où $\tilde{F}$ admet des points fixes. Nous obtenons alors un feuilletage topologique singulier transverse à la dynamique dont les singularités sont un ensemble fermé de points fixes de $F$.

A (quadratic) Hubbard tree is an invariant tree connecting the critical orbit within the Julia set of a postcritically finite (quadratic) polynomial. It is easy to read off the kneading sequences from a quadratic Hubbard tree; the result in this paper handles the converse direction. Not every sequence on two symbols is realized as the kneading sequence of a real or complex quadratic polynomial. Milnor and Thurston classified all real-admissible sequences, and we give a classification of all complex-admissible...