scholarly journals SELF-SIMILAR INVERSE SEMIGROUPS AND SMALE SPACES

2006 ◽  
Vol 16 (05) ◽  
pp. 849-874 ◽  
Author(s):  
VOLODYMYR NEKRASHEVYCH
Keyword(s):  
Markov Partition ◽  
Inverse Semigroups ◽  
Automata Theory ◽  
Markov Partitions ◽  
Self Similar ◽  
Smale Space ◽  
Smale Spaces

Self-similar inverse semigroups are defined using automata theory. Adjacency semigroups of s-resolved Markov partitions of Smale spaces are introduced. It is proved that a Smale space can be reconstructed from the adjacency semigroup of its Markov partition, using the notion of the limit solenoid of a contracting self-similar semigroup. The notions of the limit solenoid and a contracting semigroup is described.

scholarly journals Self-Similar Inverse Semigroups from Wieler Solenoids

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 266
Author(s):  
Inhyeop Yi
Keyword(s):  
Inverse Limit ◽  
Inverse Semigroups ◽  
Stable Sets ◽  
Limit System ◽  
Tracial State ◽  
C Algebras ◽  
Self Similar ◽  
Smale Space ◽  
Factor Maps ◽  
Totally Disconnected

Wieler showed that every irreducible Smale space with totally disconnected local stable sets is an inverse limit system, called a Wieler solenoid. We study self-similar inverse semigroups defined by s-resolving factor maps of Wieler solenoids. We show that the groupoids of germs and the tight groupoids of these inverse semigroups are equivalent to the unstable groupoids of Wieler solenoids. We also show that the C ∗ -algebras of the groupoids of germs have a unique tracial state.

Markov partitions and homology for -solenoids

2015 ◽  
Vol 37 (3) ◽  
pp. 716-738 ◽  
Author(s):  
NIGEL D. BURKE ◽  
IAN F. PUTNAM
Keyword(s):  
Dynamical System ◽  
Homology Theory ◽  
Prime Pair ◽  
Topological Dynamical System ◽  
Markov Partitions ◽  
Local Coordinates ◽  
Smale Space ◽  
The Given ◽  
Smale Spaces ◽  
Special Case

Given a relatively prime pair of integers, $n\geq m>1$, there is associated a topological dynamical system which we refer to as an $n/m$-solenoid. It is also a Smale space, as defined by David Ruelle, meaning that it has local coordinates of contracting and expanding directions. In this case, these are locally products of the real and various $p$-adic numbers. In the special case, $m=2,n=3$ and for $n>3m$, we construct Markov partitions for such systems. The second author has developed a homology theory for Smale spaces and we compute this in these examples, using the given Markov partitions, for all values of $n\geq m>1$ and relatively prime.

scholarly journals Symbolic dynamics for non-uniformly hyperbolic systems

2020 ◽  
pp. 1-68
Author(s):  
YURI LIMA
Keyword(s):  
Symbolic Dynamics ◽  
Hyperbolic Systems ◽  
Markov Partition ◽  
Equilibrium Measures ◽  
Markov Partitions ◽  
Recent Advances ◽  
Closed Orbits ◽  
Symbolic Extension ◽  
Uniformly Hyperbolic Systems

Abstract This survey describes the recent advances in the construction of Markov partitions for non-uniformly hyperbolic systems. One important feature of this development comes from a finer theory of non-uniformly hyperbolic systems, which we also describe. The Markov partition defines a symbolic extension that is finite-to-one and onto a non-uniformly hyperbolic locus, and this provides dynamical and statistical consequences such as estimates on the number of closed orbits and properties of equilibrium measures. The class of systems includes diffeomorphisms, flows, and maps with singularities.

scholarly journals Smale spaces via inverse limits

2013 ◽  
Vol 34 (6) ◽  
pp. 2066-2092 ◽  
Author(s):  
SUSANA WIELER
Keyword(s):  
Dynamical System ◽  
Inverse Limits ◽  
Canonical Coordinates ◽  
Chaotic Dynamical System ◽  
Smale Space ◽  
Smale Spaces ◽  
Special Case ◽  
Basic Sets

AbstractA Smale space is a chaotic dynamical system with canonical coordinates of contracting and expanding directions. The basic sets for Smale’s Axiom $A$ systems are a key class of examples. We consider the special case of irreducible Smale spaces with zero-dimensional contracting directions, and characterize these as stationary inverse limits satisfying certain conditions.

scholarly journals Continuous Orbit Equivalence on Self-Similar Graph Actions

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 990
Author(s):  
Inhyeop Yi
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Orbit Equivalence ◽  
Semigroup Actions ◽  
Infinite Path ◽  
Self Similar

For self-similar graph actions, we show that isomorphic inverse semigroups associated to a self-similar graph action are a complete invariant for the continuous orbit equivalence of inverse semigroup actions on infinite path spaces.

scholarly journals Generating special Markov partitions for hyperbolic toral automorphisms using fractals

1986 ◽  
Vol 6 (3) ◽  
pp. 325-333 ◽  
Author(s):  
Tim Bedford
Keyword(s):  
Transition Matrix ◽  
Markov Partition ◽  
Natural Conditions ◽  
Markov Partitions ◽  
Toral Automorphisms

AbstractWe show that given some natural conditions on a 3 × 3 hyperbolic matrix of integers A(det A = 1) there exists a Markov partition for the induced map A(x + ℤ3) = A(x)+ℤ3 on T3 whose transition matrix is (A−1)t. For expanding endomorphisms of T2 we construct a Markov partition so that there is a semiconjugacy from a full (one-sided) shift.

scholarly journals Groupoids associated to inverse semigroups

2021 ◽  
Author(s):  
Malcolm Jones
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Topological Isomorphism ◽  
Similar Group ◽  
Self Similar

<p>Starting with an arbitrary inverse semigroup with zero, we study two well-known groupoid constructions, yielding groupoids of filters and groupoids of germs. The groupoids are endowed with topologies making them étale. We use the bisections of the étale groupoids to show there is a topological isomorphism between the groupoids. This demonstrates a widely useful equivalence between filters and germs. We use the isomorphism to characterise Exel’s tight groupoid of germs as a groupoid of filters, to find a nice basis for the topology on the groupoid of ultrafilters and to describe the ultrafilters in the inverse semigroup of an arbitrary self-similar group.</p>

scholarly journals Ring and module structures on dimension groups associated with a shift of finite type

2012 ◽  
Vol 32 (4) ◽  
pp. 1370-1399 ◽  
Author(s):  
D. B. KILLOUGH ◽  
I. F. PUTNAM
Keyword(s):  
Finite Type ◽  
Inductive Limits ◽  
Shift Of Finite Type ◽  
Dimension Groups ◽  
Special Cases ◽  
Free Abelian Groups ◽  
Smale Space ◽  
Module Structures ◽  
Smale Spaces ◽  
Special Case

AbstractWe study invariants for shifts of finite type obtained as the K-theory of various C*-algebras associated with them. These invariants have been studied intensively over the past thirty years since their introduction by Wolfgang Krieger. They may be given quite concrete descriptions as inductive limits of simplicially ordered free abelian groups. Shifts of finite type are special cases of Smale spaces and, in earlier work, the second author has shown that the hyperbolic structure of the dynamics in a Smale space induces natural ring and module structures on certain of these K-groups. Here, we restrict our attention to the special case of shifts of finite type and obtain explicit descriptions in terms of the inductive limits.

scholarly journals Asymptotic Continuous Orbit Equivalence of Smale Spaces and Ruelle Algebras

2019 ◽  
Vol 71 (5) ◽  
pp. 1243-1296
Author(s):  
Kengo Matsumoto
Keyword(s):  
Fixed Point ◽  
Circle Action ◽  
Orbit Equivalence ◽  
Extended Version ◽  
Fixed Point Algebra ◽  
Smale Space ◽  
Smale Spaces

AbstractIn the first part of the paper, we introduce notions of asymptotic continuous orbit equivalence and asymptotic conjugacy in Smale spaces and characterize them in terms of their asymptotic Ruelle algebras with their dual actions. In the second part, we introduce a groupoid$C^{\ast }$-algebra that is an extended version of the asymptotic Ruelle algebra from a Smale space and study the extended Ruelle algebras from the view points of Cuntz–Krieger algebras. As a result, the asymptotic Ruelle algebra is realized as a fixed point algebra of the extended Ruelle algebra under certain circle action.

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