We introduce a new equivalence relation on the set of all polygonal billiards. We say that two billiards (or polygons) are order equivalent if each of the billiards has an orbit whose footpoints are dense in the boundary and the two sequences of footpoints of these orbits have the same combinatorial order. We study this equivalence relation under additional regularity conditions on the orbit.

We prove that if the Cantor set K, dynamically defined by a function $S\in C^{1+\alpha }$, satisfies the conditions of McDuff’s conjecture then it cannot be C¹-minimal.

Given the plane triangle with vertices (0,0), (0,4) and (4,0) and the transformation F: (x,y) ↦ (x(4-x-y),xy) introduced by A. N. Sharkovskiĭ, we prove the existence of the following objects: a unique invariant curve of spiral type, a periodic trajectory of period 4 (given explicitly) and a periodic trajectory of period 5 (described approximately). Also, we give a decomposition of the triangle which helps to understand the global dynamics of this discrete system which is linked with the behavior...

We study a certain class of weakly order preserving, non-invertible circle maps with irrational rotation numbers and exactly one flat interval. For this class of circle maps we explain the geometric and dynamic structure of orbits. In particular, we formulate the so called upper and lower scaling rules which show an asymmetric and double exponential decay of geometry.

We prove complex bounds for infinitely renormalizable real quadratic maps with essentially bounded combinatorics. This is the last missing ingredient in the problem of complex bounds for all infinitely renormalizable real quadratics. One of the corollaries is that the Julia set of any real quadratic map $z\mapsto z^2+c$, $c\in [-2,1/4]$, is locally connected.

It is well known that the Hubbard tree of a postcritically finite complex polynomial contains all the combinatorial information on the polynomial. In fact, an abstract Hubbard tree as defined in [23] uniquely determines the polynomial up to affine conjugation. In this paper we give necessary and sufficient conditions enabling one to deduce directly from the restriction of a quadratic Misiurewicz polynomial to its Hubbard tree whether the polynomial is renormalizable, and in this case, of which type....

Dans cet article, nous étudions la dynamique des échanges d’intervalles affines dont les pentes sont des puissances d’un même entier $m$ et dont les coupures et leurs images sont des rationnels. Nous montrons qu’une telle application a une dynamique très simple  : toutes ses orbites sont propres et elle possède au moins une orbite périodique ou un cycle périodique. Comme corollaire de ce résultat, nous montrons que les éléments de distortion dans les groupes de Higman-Thompson $V_{r,m}$ sont ceux d’ordre...

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 $t=1$.