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Shadow trees of Mandelbrot sets

Virpi Kauko (2003)

Fundamenta Mathematicae

The topology and combinatorial structure of the Mandelbrot set ℳ d (of degree d ≥ 2) can be studied using symbolic dynamics. Each parameter is mapped to a kneading sequence, or equivalently, an internal address; but not every such sequence is realized by a parameter in ℳ d . Thus the abstract Mandelbrot set is a subspace of a larger, partially ordered symbol space, Λ d . In this paper we find an algorithm to construct “visible trees” from symbolic sequences which works whether or not the sequence is realized....

Shadowing and internal chain transitivity

Jonathan Meddaugh, Brian E. Raines (2013)

Fundamenta Mathematicae

The main result of this paper is that a map f: X → X which has shadowing and for which the space of ω-limits sets is closed in the Hausdorff topology has the property that a set A ⊆ X is an ω-limit set if and only if it is closed and internally chain transitive. Moreover, a map which has the property that every closed internally chain transitive set is an ω-limit set must also have the property that the space of ω-limit sets is closed. As consequences of this result, we show that interval maps with...

Shadowing in multi-dimensional shift spaces

Piotr Oprocha (2008)

Colloquium Mathematicae

We show that the class of expansive ℤ d actions with P.O.T.P. is wider than the class of actions topologically hyperbolic in some direction ν ∈ ℤ d . Our main tool is an extension of a result by Walters to the multi-dimensional symbolic dynamics case.

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.

Subcontinua of inverse limit spaces of unimodal maps

Karen Brucks, Henk Bruin (1999)

Fundamenta Mathematicae

We discuss the inverse limit spaces of unimodal interval maps as topological spaces. Based on the combinatorial properties of the unimodal maps, properties of the subcontinua of the inverse limit spaces are studied. Among other results, we give combinatorial conditions for an inverse limit space to have only arc+ray subcontinua as proper (non-trivial) subcontinua. Also, maps are constructed whose inverse limit spaces have the inverse limit spaces of a prescribed set of periodic unimodal maps as...

Substitution dynamical systems on infinite alphabets

Sébastien Ferenczi (2006)

Annales de l’institut Fourier

We give a few examples of substitutions on infinite alphabets, and the beginning of a general theory of the associated dynamical systems. In particular, the “drunken man” substitution can be associated to an ergodic infinite measure preserving system, of Krengel entropy zero, while substitutions of constant length with a positive recurrent infinite matrix correspond to ergodic finite measure preserving systems.

Substitution systems associated with the dynamical system (𝒜, Tf)*

Maria de Fátima Correia, Carlos Ramos, Sandra Vinagre (2012)

ESAIM: Proceedings

We consider the dynamical system (𝒜, Tf), where 𝒜 is a class of differential real functions defined on some interval and Tf : 𝒜 → 𝒜 is an operator Tfφ := fοφ, where f is a differentiable m-modal map. If we consider functions in 𝒜 whose critical values are periodic points for f then, we show how to define and characterize a substitution system associated with (𝒜, Tf). For these substitution systems, we compute the growth rate of the...

Substitutions, abstract number systems and the space filling property

Clemens Fuchs, Robert Tijdeman (2006)

Annales de l’institut Fourier

In this paper we study multi-dimensional words generated by fixed points of substitutions by projecting the integer points on the corresponding broken halfline. We show for a large class of substitutions that the resulting word is the restriction of a linear function modulo 1 and that it can be decided whether the resulting word is space filling or not. The proof uses lattices and the abstract number system associated with the substitution.

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.

Substitutions par des motifs en dimension 1

N. Pytheas Fogg (2007)

RAIRO - Theoretical Informatics and Applications

Une substitution est un morphisme de monoïdes libres : chaque lettre a pour image un mot, et l'image d'un mot est la concaténation des images de ses lettres. Cet article introduit une généralisation de la notion de substitution, où l'image d'une lettre n'est plus un mot mais un motif, c'est-à-dire un “mot à trous”, l'image d'un mot étant obtenue en raccordant les motifs correspondant à chacune de ses lettres à l'aide de règles locales. On caractérise complètement les substitutions par des motifs...

Suites doubles de basse complexité

Valérie Berthé, Laurent Vuillon (2000)

Journal de théorie des nombres de Bordeaux

Nous donnons une représentation géométrique des suites doubles uniformément récurrentes de fonction de complexité rectangulaire m n + n . Nous montrons que ces suites codent l’action d’une ℤ 2 -action définie par deux rotations irrationnelles sur le cercle unité. La preuve repose sur une étude des suites doubles dont les lignes sont des suite sturmiennes de même langage.

Sur le codage du flot géodésique dans un arbre

Anne Broise-Alamichel, Frédéric Paulin (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

Étant donné un arbre T et un groupe Γ d’automorphismes de T , nous étudions les propriétés markoviennes du flot géodésique sur le quotient de l’espace des géodésiques de T par Γ . Par exemple, quand T est l’arbre de Bruhat-Tits d’un groupe algébrique linéaire connexe semi-simple G ̲ de rang 1 sur un corps local non archimédien K ^ et si Γ est un réseau (éventuellement non uniforme) dans G ̲ ( K ^ ) , nous montrons que l’action des puissances paires de la transformation géodésique est Bernoulli d’entropie finie sur...

Sur les processus quasi-Markoviens et certains de leurs facteurs

Thierry de la Rue (2005)

Colloquium Mathematicae

We study a class of stationary finite state processes, called quasi-Markovian, including in particular the processes whose law is a Gibbs measure as defined by Bowen. We show that, if a factor with integrable coding time of a quasi-Markovian process is maximal in entropy, then this factor splits off, which means that it admits a Bernoulli shift as an independent complement. If it is not maximal in entropy, then we can find a splitting finite extension of this factor, which generalizes a theorem...

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