scholarly journals A note on homology for Smale spaces

2020 ◽  
Vol 14 (3) ◽  
pp. 813-836
Author(s):  
Valerio Proietti
Keyword(s):  
Smale Spaces

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.

Order on the homology groups of Smale spaces

2017 ◽  
Vol 288 (2) ◽  
pp. 257-288
Author(s):  
Massoud Amini ◽  
Ian Putnam ◽  
Sarah Saeidi Gholikandi
Keyword(s):  
Homology Groups ◽  
Smale Spaces

scholarly journals Homology for one-dimensional solenoids

2017 ◽  
Vol 121 (2) ◽  
pp. 219 ◽  
Author(s):  
Massoud Amini ◽  
Ian F. Putnam ◽  
Sarah Saeidi Gholikandi
Keyword(s):  
Finite Type ◽  
Homology Theory ◽  
One Dimensional ◽  
Axiom A ◽  
Group Invariant ◽  
Dimension Group ◽  
Basic Set ◽  
The One ◽  
Topological Dynamical Systems ◽  
Smale Spaces

Smale spaces are a particular class of hyperbolic topological dynamical systems, defined by David Ruelle. The definition was introduced to give an axiomatic description of the dynamical properties of Smale's Axiom A systems when restricted to a basic set. They include Anosov diffeomeorphisms, shifts of finite type and various solenoids constructed by R. F. Williams. The second author constructed a homology theory for Smale spaces which is based on (and extends) Krieger's dimension group invariant for shifts of finite type. In this paper, we compute this homology for the one-dimensional generalized solenoids of R. F. Williams.

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.

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 A note on the ideals of groupoid C*-algebras from Smale spaces

2001 ◽  
Vol 64 (2) ◽  
pp. 271-279 ◽  
Author(s):  
Chengjun Hou ◽  
Xiamoman Chen
Keyword(s):  
Equivalence Relation ◽  
Asymptotic Equivalence ◽  
Topologically Transitive ◽  
C Algebras ◽  
Smale Space ◽  
Smale Spaces

In this note, we characterise completely the ideals of the groupoid C*-algebra arising from the asymptotic equivalence relation on the points of a Smale space and show that the related Ruelle algebra is simple when the Smale space is topologically transitive.

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