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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.
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.
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 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.
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.
We give analogs of the complexity and of Sturmian words which are called respectively the -complexity and -Sturmian words. We show that the class of -Sturmian words coincides with the class of words satisfying , and we determine the structure of -Sturmian words. For a class of words satisfying , we give a general formula and an upper bound for . Using this general formula, we give explicit formulae for for some words belonging to this class. In general, can take large values, namely,...
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...
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.
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...
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 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.
L’étude des systèmes dynamiques non archimédiens initiée par J. Lubin conduit à déterminer la ramification de séries à coefficients dans un corps fini , qui commutent entre elles pour la loi . Dans cet article nous traitons le cas des sous-groupes abéliens de qui correspondent par le foncteur corps de normes aux extensions abéliennes des extensions finies de , dont la ramification se stabilise dès le début.
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.
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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