scholarly journals Bowen Measure From Heteroclinic Points

2012 ◽  
Vol 64 (6) ◽  
pp. 1341-1358 ◽  
Author(s):  
D. B. Killough ◽  
I. F. Putnam
Keyword(s):  
Growth Rate ◽  
Invariant Probability Measure ◽  
Periodic Points ◽  
Direct Computation ◽  
Shift Of Finite Type ◽  
New Construction ◽  
Smale Space ◽  
Factor Maps ◽  
Smale Spaces ◽  
Heteroclinic Points

Abstract We present a new construction of the entropy-maximizing, invariant probability measure on a Smale space (the Bowen measure). Our construction is based on points that are unstably equivalent to one given point, and stably equivalent to another, i.e., heteroclinic points. The spirit of the construction is similar to Bowen's construction from periodic points, though the techniques are very different. We also prove results about the growth rate of certain sets of heteroclinic points, and about the stable and unstable components of the Bowen measure. The approach we take is to prove results through direct computation for the case of a Shift of Finite type, and then use resolving factor maps to extend the results to more general Smale spaces.

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 On the growth rate inequality for periodic points in the two sphere

2019 ◽  
Vol 25 (2) ◽  
pp. 219-232
Author(s):  
Gerardo Honorato ◽  
Jorge Iglesias ◽  
Aldo Portela ◽  
Alvaro Rovella ◽  
Francisco Valenzuela ◽  
...  
Keyword(s):  
Growth Rate ◽  
Periodic Points

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.

The Optimum Timing and Maximum Impact of Full Rehabilitation of New Zealand Housing Stock

1998 ◽  
Vol 30 (7) ◽  
pp. 1295-1311 ◽  
Author(s):  
I M Johnstone
Keyword(s):  
Growth Rate ◽  
Population Dynamics ◽  
New Zealand ◽  
Housing Stock ◽  
Maintenance Costs ◽  
Total Cost ◽  
Empirical Estimates ◽  
Stable Housing ◽  
New Construction ◽  
The Cost

The author develops a simulation model to estimate the optimum timing and maximum impact of full rehabilitation of New Zealand housing stock. The model is based on the theories of classical population dynamics. Data used in the model include empirical estimates of the mortality of New Zealand housing stock, assumed schedules of depreciation of dwelling services, and assumed schedules of annual maintenance costs. The dwelling service years provided by dwellings serve as a proxy for benefits of rents or imputed rents (excluding rent for land). The cost to construct one dwelling and fractions thereof serve as a proxy for costs of maintenance, rehabilitation, replacement, and new construction. Optimum timing of rehabilitation can increase the quantity of benefits provided by the housing stock per unit total cost but a reduction in the growth rate of new dwellings has a greater impact in achieving the same objective. A stationary and stable housing stock can provide 45% more dwelling services per unit total cost than a housing stock which doubles in size every 35 years.

scholarly journals Dynamical properties of some adic systems with arbitrary orderings

2016 ◽  
Vol 37 (7) ◽  
pp. 2131-2162 ◽  
Author(s):  
SARAH FRICK ◽  
KARL PETERSEN ◽  
SANDI SHIELDS
Keyword(s):  
Probability Measure ◽  
Fixed Points ◽  
Invariant Probability Measure ◽  
Random Order ◽  
Periodic Points ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Weakly Mixing ◽  
Dynamical Properties ◽  
Necessary And Sufficient

We consider arbitrary orderings of the edges entering each vertex of the (downward directed) Pascal graph. Each ordering determines an adic (Bratteli–Vershik) system, with a transformation that is defined on most of the space of infinite paths that begin at the root. We prove that for every ordering the coding of orbits according to the partition of the path space determined by the first three edges is essentially faithful, meaning that it is one-to-one on a set of paths that has full measure for every fully supported invariant probability measure. We also show that for every$k$the subshift that arises from coding orbits according to the first$k$edges is topologically weakly mixing. We give a necessary and sufficient condition for any adic system to be topologically conjugate to an odometer and use this condition to determine the probability that a random order on a fixed diagram, or a diagram constructed at random in some way, is topologically conjugate to an odometer. We also show that the closure of the union over all orderings of the subshifts arising from codings of the Pascal adic by the first edge has superpolynomial complexity, is not topologically transitive, and has no periodic points besides the two fixed points, while the intersection over all orderings consists of just four orbits.

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.

Lattice invariants for sofic shifts

1991 ◽  
Vol 11 (4) ◽  
pp. 787-801 ◽  
Author(s):  
Susan Williams
Keyword(s):  
Finite Type ◽  
Necessary Conditions ◽  
Shift Of Finite Type ◽  
Sofic Shift ◽  
Dimension Group ◽  
Sofic Shifts ◽  
Factor Map ◽  
Factor Maps

AbstractTo a factor map φ from an irreducible shift of finite type ΣAto a sofic shiftS, we associate a subgroup of the dimension group (GA, Â) which is an invariant of eventual conjugacy for φ. This invariant yields new necessary conditions for the existence of factor maps between equal entropy sofic shifts.

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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