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Some new examples of recurrence and non-recurrence sets for products of rotations on the unit circle

Sophie Grivaux, Maria Roginskaya (2013)

Czechoslovak Mathematical Journal

We study recurrence and non-recurrence sets for dynamical systems on compact spaces, in particular for products of rotations on the unit circle 𝕋 . A set of integers is called r -Bohr if it is recurrent for all products of r rotations on 𝕋 , and Bohr if it is recurrent for all products of rotations on 𝕋 . It is a result due to Katznelson that for each r ≥ 1 there exist sets of integers which are r -Bohr but not ( r + 1 ) -Bohr. We present new examples of r -Bohr sets which are not Bohr, thanks to a construction which...

Spaces of ω-limit sets of graph maps

Jie-Hua Mai, Song Shao (2007)

Fundamenta Mathematicae

Let (X,f) be a dynamical system. In general the set of all ω-limit sets of f is not closed in the hyperspace of closed subsets of X. In this paper we study the case when X is a graph, and show that the family of ω-limit sets of a graph map is closed with respect to the Hausdorff metric.

Statistical properties of unimodal maps

Artur Avila, Carlos Gustavo Moreira (2005)

Publications Mathématiques de l'IHÉS

We consider typical analytic unimodal maps which possess a chaotic attractor. Our main result is an explicit combinatorial formula for the exponents of periodic orbits. Since the exponents of periodic orbits form a complete set of smooth invariants, the smooth structure is completely determined by purely topological data (“typical rigidity”), which is quite unexpected in this setting. It implies in particular that the lamination structure of spaces of analytic unimodal maps (obtained by the partition...

Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles

Claire Chavaudret (2013)

Bulletin de la Société Mathématique de France

This article is about almost reducibility of quasi-periodic cocycles with a diophantine frequency which are sufficiently close to a constant. Generalizing previous works by L.H. Eliasson, we show a strong version of almost reducibility for analytic and Gevrey cocycles, that is to say, almost reducibility where the change of variables is in an analytic or Gevrey class which is independent of how close to a constant the initial cocycle is conjugated. This implies a result of density, or quasi-density,...

Strong Transitivity and Graph Maps

Katsuya Yokoi (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

We study the relation between transitivity and strong transitivity, introduced by W. Parry, for graph self-maps. We establish that if a graph self-map f is transitive and the set of fixed points of f k is finite for each k ≥ 1, then f is strongly transitive. As a corollary, if a piecewise monotone graph self-map is transitive, then it is strongly transitive.

Structure of inverse limit spaces of tent maps with finite critical orbit

Sonja Štimac (2006)

Fundamenta Mathematicae

Using methods of symbolic dynamics, we analyze the structure of composants of the inverse limit spaces of tent maps with finite critical orbit. We define certain symmetric arcs called bridges. They are building blocks of composants. Then we show that the folding patterns of bridges are characterized by bridge types and prove that there are finitely many bridge types.

Substitutions on two letters, cutting segments and their projections

Sierk W. Rosema (2007)

Journal de Théorie des Nombres de Bordeaux

In this paper we study the structure of the projections of the finite cutting segments corresponding to unimodular substitutions over a two-letter alphabet. We show that such a projection is a block of letters if and only if the substitution is Sturmian. Applying the procedure of projecting the cutting segments corresponding to a Christoffel substitution twice results in the original substitution. This induces a duality on the set of Christoffel substitutions.

Sulla stabilità di un punto fisso per funzioni di n variabili complesse. Problema del Centro di Schröder-Siegel

Timoteo Carletti (2005)

Bollettino dell'Unione Matematica Italiana

Viene considerato il problema della stabilità di un punto fisso per un germe di diffeomorfismo di più variabili complesse cercando un coniugio con la sua parte lineare: Problema del centro di Schröder-Siegel. Dopo aver formulato il problema e ricordato i principali risultati nel caso di diffeomorfismi olomorfi, mostriamo come estendere il problema ad alcune situazioni non olomorfe, in particolare ci interesseremo al caso di germi Gevrey. Concluderemo con un'applicazione rivolta a mostrare la stabilità...

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