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 Structure of transition classes for factor codes on shifts of finite type

2014 ◽  
Vol 35 (8) ◽  
pp. 2353-2370 ◽  
Author(s):  
MAHSA ALLAHBAKHSHI ◽  
SOONJO HONG ◽  
UIJIN JUNG
Keyword(s):  
Finite Type ◽  
Irreducible Factor ◽  
Shift Of Finite Type ◽  
Sofic Shift ◽  
Transitive Point ◽  
Dynamical Properties ◽  
Factor Code

Given a factor code ${\it\pi}$ from a shift of finite type $X$ onto a sofic shift $Y$, the class degree of ${\it\pi}$ is defined to be the minimal number of transition classes over the points of $Y$. In this paper, we investigate the structure of transition classes and present several dynamical properties analogous to the properties of fibers of finite-to-one factor codes. As a corollary, we show that for an irreducible factor triple, there cannot be a transition between two distinct transition classes over a right transitive point, answering a question raised by Quas.

scholarly journals Compensation functions for factors of shifts of finite type

2014 ◽  
Vol 36 (2) ◽  
pp. 375-389 ◽  
Author(s):  
JOHN ANTONIOLI
Keyword(s):  
Continuous Function ◽  
Invariant Measure ◽  
Finite Type ◽  
Relative Equilibrium ◽  
Equilibrium States ◽  
Shift Of Finite Type ◽  
Dini Condition ◽  
Factor Map ◽  
Compensation Function ◽  
Summable Variation

Let ${\it\pi}:X\rightarrow Y$ be an infinite-to-one factor map, where $X$ is a shift of finite type. A compensation function relates equilibrium states on $X$ to equilibrium states on $Y$. The $p$-Dini condition is given as a way of measuring the smoothness of a continuous function, with $1$-Dini corresponding to functions with summable variation. Two types of compensation functions are defined in terms of this condition. Given a fully supported invariant measure ${\it\nu}$ on $Y$, we show that the relative equilibrium states of a $1$-Dini function $f$ over ${\it\nu}$ are themselves fully supported, and have positive relative entropy. We then show that there exists a compensation function which is $p$-Dini for all $p>1$ which has relative equilibrium states supported by a subshift on which ${\it\pi}$ is a finite-to-one map onto $Y$.

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 Normal amenable subgroups of the automorphism group of sofic shifts

2020 ◽  
pp. 1-14
Author(s):  
KITTY YANG
Keyword(s):  
Automorphism Group ◽  
Finite Type ◽  
Higher Dimensions ◽  
Two Dimensional ◽  
Shift Of Finite Type ◽  
Sofic Shift ◽  
Sofic Shifts ◽  
Amenable Subgroup

Let $(X,\unicode[STIX]{x1D70E})$ be a transitive sofic shift and let $\operatorname{Aut}(X)$ denote its automorphism group. We generalize a result of Frisch, Schlank, and Tamuz to show that any normal amenable subgroup of $\operatorname{Aut}(X)$ must be contained in the subgroup generated by the shift. We also show that the result does not extend to higher dimensions by giving an example of a two-dimensional mixing shift of finite type due to Hochman whose automorphism group is amenable and not generated by the shift maps.

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 Decompositions of factor codes and embeddings between shift spaces with unequal entropies

2012 ◽  
Vol 33 (1) ◽  
pp. 144-157
Author(s):  
SOONJO HONG ◽  
UIJIN JUNG ◽  
IN-JE LEE
Keyword(s):  
Finite Type ◽  
Shift Spaces ◽  
Sofic Shift ◽  
Shift Space ◽  
Sofic Shifts ◽  
The Given ◽  
Factor Code

AbstractGiven a factor code between sofic shifts X and Y, there is a family of decompositions of the original code into factor codes such that the entropies of the intermediate subshifts arising from the decompositions are dense in the interval from the entropy of Y to that of X. Furthermore, if X is of finite type, we can choose those intermediate subshifts as shifts of finite type. In the second part of the paper, given an embedding from a shift space to an irreducible sofic shift, we characterize the set of the entropies of the intermediate subshifts arising from the decompositions of the given embedding into embeddings.

scholarly journals Existence of a measurable saturated compensation function between subshifts and its applications

2010 ◽  
Vol 31 (5) ◽  
pp. 1563-1589 ◽  
Author(s):  
YUKI YAYAMA
Keyword(s):  
Hausdorff Dimension ◽  
Finite Type ◽  
Symbolic Representation ◽  
Invariant Set ◽  
Conformal Map ◽  
Constant Multiple ◽  
Shift Of Finite Type ◽  
Ergodic Measures ◽  
Borel Measurable ◽  
Compensation Function

AbstractWe show the existence of a bounded Borel measurable saturated compensation function for any factor map between subshifts. As an application, we find the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set of an expanding non-conformal map on the torus given by an integer-valued diagonal matrix. These problems were studied in [23] for a compact invariant set whose symbolic representation is a shift of finite type under the condition of the existence of a saturated compensation function. By using the ergodic equilibrium states of a constant multiple of a Borel measurable compensation function, we extend the results to the general case where this condition might not hold, presenting a formula for the Hausdorff dimension for a compact invariant set whose symbolic representation is a subshift and studying invariant ergodic measures of full dimension. We study uniqueness and properties of such measures for a compact invariant set whose symbolic representation is a topologically mixing shift of finite type.

scholarly journals A characterization of ω-limit sets in shift spaces

2009 ◽  
Vol 30 (1) ◽  
pp. 21-31 ◽  
Author(s):  
ANDREW BARWELL ◽  
CHRIS GOOD ◽  
ROBIN KNIGHT ◽  
BRIAN E. RAINES
Keyword(s):  
Finite Type ◽  
Limit Set ◽  
Invariant Subset ◽  
Limit Sets ◽  
Tent Map ◽  
Shift Of Finite Type ◽  
Sofic Shift ◽  
Shift Space ◽  
Full Shift

AbstractA set Λ is internally chain transitive if for any x,y∈Λ and ϵ>0 there is an ϵ-pseudo-orbit in Λ between x and y. In this paper we characterize all ω-limit sets in shifts of finite type by showing that, if Λ is a closed, strongly shift-invariant subset of a shift of finite type, X, then there is a point z∈X with ω(z)=Λ if and only if Λ is internally chain transitive. It follows immediately that any closed, strongly shift-invariant, internally chain transitive subset of a shift space over some alphabet ℬ is the ω-limit set of some point in the full shift space over ℬ. We use similar techniques to prove that, for a tent map f, a closed, strongly f-invariant, internally chain transitive subset of the interval is the ω-limit set of a point provided it does not contain the image of the critical point. We give an example of a sofic shift space Z𝒢 (a factor of a shift space of finite type) that is not of finite type that has an internally chain transitive subset that is not the ω-limit set of any point in Z𝒢.

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.

scholarly journals A Study on f Number of Points for Number of Points with Inter-Distance of Less Than, or for Number of Points with Inter-Distance of or More to Necessarily Exist on Plane

2018 ◽  
Author(s):  
Benjamin Smith
Keyword(s):  
Upper Bound ◽  
Ramsey Number ◽  
Minimum Value ◽  
Special Case

We defined number of points with an inter-distance of β or more to necessarily exist on a plane. Furthermore, we aimed to reduce the range of this minimum value. We first showed that the upper bound of this value could be scaled by , and further reduced the constant that was multiplied. We compared the upper bound of and the Ramsey number in a special case and confirmed that was a better upper bound than except when were both small or trivial.

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