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Stable rank and real rank of compact transformation group C*-algebras

Robert J. Archbold, Eberhard Kaniuth (2006)

Studia Mathematica

Let (G,X) be a transformation group, where X is a locally compact Hausdorff space and G is a compact group. We investigate the stable rank and the real rank of the transformation group C*-algebra C₀(X)⋊ G. Explicit formulae are given in the case where X and G are second countable and X is locally of finite G-orbit type. As a consequence, we calculate the ranks of the group C*-algebra C*(ℝⁿ ⋊ G), where G is a connected closed subgroup of SO(n) acting on ℝⁿ by rotation.

Stretching the Oxtoby-Ulam Theorem

Ethan Akin (2000)

Colloquium Mathematicae

On a manifold X of dimension at least two, let μ be a nonatomic measure of full support with μ(∂X) = 0. The Oxtoby-Ulam Theorem says that ergodicity of μ is a residual property in the group of homeomorphisms which preserve μ. Daalderop and Fokkink have recently shown that density of periodic points is residual as well. We provide a proof of their result which replaces the dependence upon the Annulus Theorem by a direct construction which assures topologically robust periodic points.

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.

*-sturmian words and complexity

Izumi Nakashima, Jun-Ichi Tamura, Shin-Ichi Yasutomi (2003)

Journal de théorie des nombres de Bordeaux

We give analogs of the complexity p ( n ) and of Sturmian words which are called respectively the * -complexity p * ( n ) and * -Sturmian words. We show that the class of * -Sturmian words coincides with the class of words satisfying p * ( n ) ≤ n + 1 , and we determine the structure of * -Sturmian words. For a class of words satisfying p * ( n ) = n + 1 , we give a general formula and an upper bound for p ( n ) . Using this general formula, we give explicit formulae for p ( n ) for some words belonging to this class. In general, p ( n ) can take large values, namely,...

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 commutatives de séries formelles

François Laubie (2000)

Journal de théorie des nombres de Bordeaux

L’étude des systèmes dynamiques non archimédiens initiée par J. Lubin conduit à déterminer la ramification de séries à coefficients dans un corps fini k , qui commutent entre elles pour la loi ∘ . Dans cet article nous traitons le cas des sous-groupes abéliens de t + t 2 k [ [ t ] ] qui correspondent par le foncteur corps de normes aux extensions abéliennes des extensions finies de ℚ p , dont la ramification se stabilise dès le début.

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...

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