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 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 Relative equilibrium states and class degree

2017 ◽  
Vol 39 (4) ◽  
pp. 865-888
Author(s):  
MAHSA ALLAHBAKHSHI ◽  
JOHN ANTONIOLI ◽  
JISANG YOO
Keyword(s):  
Finite Type ◽  
Upper Bound ◽  
Relative Equilibrium ◽  
Ergodic Measure ◽  
Equilibrium States ◽  
Shift Of Finite Type ◽  
Sofic Shift ◽  
Ergodic Measures ◽  
Factor Code ◽  
Special Case

Given a factor code $\unicode[STIX]{x1D70B}$ from a shift of finite type $X$ onto a sofic shift $Y$, an ergodic measure $\unicode[STIX]{x1D708}$ on $Y$, and a function $V$ on $X$ with sufficient regularity, we prove an invariant upper bound on the number of ergodic measures on $X$ which project to $\unicode[STIX]{x1D708}$ and maximize the measure pressure $h(\unicode[STIX]{x1D707})+\int V\,d\unicode[STIX]{x1D707}$ among all measures in the fiber $\unicode[STIX]{x1D70B}^{-1}(\unicode[STIX]{x1D708})$. If $\unicode[STIX]{x1D708}$ is fully supported, this bound is the class degree of $\unicode[STIX]{x1D70B}$. This generalizes a previous result for the special case of $V=0$ and thus settles a conjecture raised by Allahbakhshi and Quas.

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.

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.

Perturbations of multidimensional shifts of finite type

2010 ◽  
Vol 31 (2) ◽  
pp. 483-526 ◽  
Author(s):  
RONNIE PAVLOV
Keyword(s):  
Finite Type ◽  
Lower Bounds ◽  
Symbolic Dynamics ◽  
Upper And Lower Bounds ◽  
Shift Of Finite Type ◽  
Strongly Irreducible ◽  
Dimensional Cube

AbstractIn this paper, we study perturbations of multidimensional shifts of finite type. Specifically, for any ℤd shift of finite type X with d>1 and any finite pattern w in the language of X, we denote by Xw the set of elements of X not containing w. For strongly irreducible X and patterns w with shape a d-dimensional cube, we obtain upper and lower bounds on htop (X)−htop (Xw) dependent on the size of w. This extends a result of Lind for d=1 . We also apply our methods to an undecidability question in ℤd symbolic dynamics.

Analytical Techniques for the Optimisation of Rocket Trajectories

1963 ◽  
Vol 14 (2) ◽  
pp. 105-124 ◽  
Author(s):  
Derek F. Lawden
Keyword(s):  
Gravitational Field ◽  
Artificial Satellite ◽  
Analytical Techniques ◽  
Present Position ◽  
Mayer Problem ◽  
Special Cases ◽  
Minimum Expenditure ◽  
Law Of Attraction ◽  
Special Case

SummaryThe development during the last two decades of analytical techniques for the solution of problems relating to the optimisation of rocket trajectories is outlined and the present position in this field of research is summarised. It is shown that the determination of optimal trajectories in a general gravitational field can be expressed as a Mayer problem from the calculus of variations. The known solution to such a problem is stated and applied, first to the special case of the launching of an artificial satellite into a circular orbit with minimum expenditure of propellant and, secondly, to the general astronautical problem of the economical transfer of a rocket between two terminals in a gravitational field. The special cases when the field is uniform and when it obeys an inverse square law of attraction to a point are then considered, and the paper concludes with some remarks concerning areas in which further investigations are necessary.

scholarly journals Water entry of an expanding wedge/plate with flow detachment

2016 ◽  
Vol 797 ◽  
pp. 322-344 ◽  
Author(s):  
Yuriy A. Semenov ◽  
Guo Xiong Wu
Keyword(s):  
Free Surface ◽  
General Solution ◽  
Water Entry ◽  
Hodograph Method ◽  
Fixed Length ◽  
Pressure Distributions ◽  
Initial Stage ◽  
Special Cases ◽  
Wedge Plate ◽  
Special Case

A general similarity solution for water-entry problems of a wedge with its inner angle fixed and its sides in expansion is obtained with flow detachment, in which the speed of expansion is a free parameter. The known solutions for a wedge of a fixed length at the initial stage of water entry without flow detachment and at the final stage corresponding to Helmholtz flow are obtained as two special cases, at some finite and zero expansion speeds, respectively. An expanding horizontal plate impacting a flat free surface is considered as the special case of the general solution for a wedge inner angle equal to ${\rm\pi}$. An initial impulse solution for a plate of a fixed length is obtained as the special case of the present formulation. The general solution is obtained in the form of integral equations using the integral hodograph method. The results are presented in terms of free-surface shapes, streamlines and pressure distributions.

scholarly journals On Reay's Relaxed Tverberg Conjecture and Generalizations of Conway's Thrackle Conjecture

10.37236/6516 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Megumi Asada ◽  
Ryan Chen ◽  
Florian Frick ◽  
Frederick Huang ◽  
Maxwell Polevy ◽  
...  
Keyword(s):  
Geometric Property ◽  
Convex Sets ◽  
Convex Hulls ◽  
Open Problems ◽  
Higher Dimensional ◽  
Special Cases ◽  
Minimum Number ◽  
Point Set ◽  
Colored Version ◽  
Special Case

Reay's relaxed Tverberg conjecture and Conway's thrackle conjecture are open problems about the geometry of pairwise intersections. Reay asked for the minimum number of points in Euclidean $d$-space that guarantees any such point set admits a partition into $r$ parts, any $k$ of whose convex hulls intersect. Here we give new and improved lower bounds for this number, which Reay conjectured to be independent of $k$. We prove a colored version of Reay's conjecture for $k$ sufficiently large, but nevertheless $k$ independent of dimension $d$. Pairwise intersecting convex hulls have severely restricted combinatorics. This is a higher-dimensional analogue of Conway's thrackle conjecture or its linear special case. We thus study convex-geometric and higher-dimensional analogues of the thrackle conjecture alongside Reay's problem and conjecture (and prove in two special cases) that the number of convex sets in the plane is bounded by the total number of vertices they involve whenever there exists a transversal set for their pairwise intersections. We thus isolate a geometric property that leads to bounds as in the thrackle conjecture. We also establish tight bounds for the number of facets of higher-dimensional analogues of linear thrackles and conjecture their continuous generalizations.

scholarly journals CUNTZ–KRIEGER ALGEBRAS AND ONE-SIDED CONJUGACY OF SHIFTS OF FINITE TYPE AND THEIR GROUPOIDS

2019 ◽  
Vol 109 (3) ◽  
pp. 289-298
Author(s):  
KEVIN AGUYAR BRIX ◽  
TOKE MEIER CARLSEN
Keyword(s):  
Conjugacy Class ◽  
Finite Type ◽  
Negative Answer ◽  
The Other ◽  
Completely Positive Map ◽  
Completely Positive ◽  
Positive Map ◽  
Shift Of Finite Type ◽  
Abelian Subalgebra ◽  
The One

AbstractA one-sided shift of finite type $(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$ determines on the one hand a Cuntz–Krieger algebra ${\mathcal{O}}_{A}$ with a distinguished abelian subalgebra ${\mathcal{D}}_{A}$ and a certain completely positive map $\unicode[STIX]{x1D70F}_{A}$ on ${\mathcal{O}}_{A}$. On the other hand, $(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$ determines a groupoid ${\mathcal{G}}_{A}$ together with a certain homomorphism $\unicode[STIX]{x1D716}_{A}$ on ${\mathcal{G}}_{A}$. We show that each of these two sets of data completely characterizes the one-sided conjugacy class of $\mathsf{X}_{A}$. This strengthens a result of Cuntz and Krieger. We also exhibit an example of two irreducible shifts of finite type which are eventually conjugate but not conjugate. This provides a negative answer to a question of Matsumoto of whether eventual conjugacy implies conjugacy.

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