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A topological characterization of holomorphic parabolic germs in the plane

Frédéric Le Roux (2008)

Fundamenta Mathematicae

J.-M. Gambaudo and É. Pécou introduced the "linking property" in the study of the dynamics of germs of planar homeomorphisms in order to provide a new proof of Naishul's theorem. In this paper we prove that the negation of the Gambaudo-Pécou property characterizes the topological dynamics of holomorphic parabolic germs. As a consequence, a rotation set for germs of surface homeomorphisms around a fixed point can be defined, and it turns out to be non-trivial except for countably many conjugacy classes....

Asymptotic period in dynamical systems in metric spaces

Karol Gryszka (2015)

Colloquium Mathematicae

We introduce the notions of asymptotic period and asymptotically periodic orbits in metric spaces. We study some properties of these notions and their connections with ω-limit sets. We also discuss the notion of growth rate of such orbits and describe its properties in an extreme case.

Does a billiard orbit determine its (polygonal) table?

Jozef Bobok, Serge Troubetzkoy (2011)

Fundamenta Mathematicae

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.

Misiurewicz maps unfold generically (even if they are critically non-finite)

Sebastian van Strien (2000)

Fundamenta Mathematicae

We show that in normalized families of polynomial or rational maps, Misiurewicz maps (critically finite or infinite) unfold generically. For example, if f λ 0 is critically finite with non-degenerate critical point c 1 ( λ 0 ) , . . . , c n ( λ 0 ) such that f λ 0 k i ( c i ( λ 0 ) ) = p i ( λ 0 ) are hyperbolic periodic points for i = 1,...,n, then  IV-1. Age impartible......................................................................................................................................................................... 31   λ ( f λ k 1 ( c 1 ( λ ) ) - p 1 ( λ ) , . . . , f λ k d - 2 ( c d - 2 ( λ ) ) - p d - 2 ( λ ) ) is a local diffeomorphism...

Nombre de rotation, mesures invariantes et ratio set des homéomorphismes affines par morceaux du cercle

Isabelle Liousse (2005)

Annales de l’institut Fourier

Etant donné α irrationnel de type constant, nous donnons des conditions explicites et génériques sur les pentes d’un homéomorphisme f affine par morceaux du cercle de nombre de rotation α , qui garantissent que la mesure de probabilité f -invariante est singulière par rapport à la mesure de Haar. Cet article contient une preuve élémentaire d’un résultat de E. Ghys et V. Sergiescu : ”le nombre de rotation d’un homéomorphisme dyadique est rationnel”. Nous y étudions aussi le ratio set des homéomorphismes...

On conjugacy equation in dimension one

Krzysztof Ciepliński, Zbigniew Leśniak (2013)

Banach Center Publications

In this paper, recent results on the existence and uniqueness of (continuous and homeomorphic) solutions φ of the equation φ ∘ f = g ∘ φ (f and g are given self-maps of an interval or the circle) are surveyed. Some applications of these results as well as the outcomes concerning systems of such equations are also presented.

On the classes of Lipschitz and smooth conjugacies of unimodal maps

Waldemar Pałuba (2004)

Fundamenta Mathematicae

Under very mild assumptions, any Lipschitz continuous conjugacy between the closures of the postcritical sets of two C¹-unimodal maps has a derivative at the critical point, and also on a dense set of its preimages. In a more restrictive situation of infinitely renormalizable maps of bounded combinatorial type the Lipschitz condition automatically implies the C¹-smoothness of the conjugacy. Here the critical degree can be any real number α > 1.

Periodicity of β-expansions for certain Pisot units*

Sandra Vaz, Pedro Martins Rodrigues (2012)

ESAIM: Proceedings

Given β > 1, let Tβ T β : [ 0 , 1 [ [ 0 , 1 [ x βx βx . The iteration of this transformation gives rise to the greedy β-expansion. There has been extensive research on the properties of this expansion and its dependence on the parameter β.In [17], K. Schmidt analyzed the set of periodic points of Tβ, where β is a Pisot number. In an attempt to generalize some of his results, we study, for certain Pisot units, a different expansion that we call linear expansion x = i 0 e i β i , where each ei...

Poincaré-Hopf index and partial hyperbolicity

C. A Morales (2008)

Annales de la faculté des sciences de Toulouse Mathématiques

We use the theory of partially hyperbolic systems [HPS] in order to find singularities of index 1 for vector fields with isolated zeroes in a 3 -ball. Indeed, we prove that such zeroes exists provided the maximal invariant set in the ball is partially hyperbolic, with volume expanding central subbundle, and the strong stable manifolds of the singularities are unknotted in the ball.

Results and open questions on some invariants measuring the dynamical complexity of a map

Jaume Llibre, Radu Saghin (2009)

Fundamenta Mathematicae

Let f be a continuous map on a compact connected Riemannian manifold M. There are several ways to measure the dynamical complexity of f and we discuss some of them. This survey contains some results and open questions about relationships between the topological entropy of f, the volume growth of f, the rate of growth of periodic points of f, some invariants related to exterior powers of the derivative of f, and several invariants measuring the topological complexity of f: the degree (for the case...

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