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 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.

RELATIVE EQUILIBRIUM STATES AND DIMENSIONS OF FIBERWISE INVARIANT MEASURES FOR RANDOM DISTANCE EXPANDING MAPS

2013 ◽  
Vol 14 (01) ◽  
pp. 1350015 ◽  
Author(s):  
DAVID SIMMONS ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Invariant Measure ◽  
Invariant Measures ◽  
Gibbs State ◽  
Relative Equilibrium ◽  
Equilibrium States ◽  
Expanding Maps ◽  
Hölder Continuous ◽  
Random Maps ◽  
Full Dimension ◽  
Relative Metric

We show that the Gibbs states (known from [10] to be unique) of Hölder continuous potentials and random distance expanding maps coincide with relative equilibrium states of those potentials, proving in particular that the latter exist and are unique. In the realm of conformal expanding random maps, we prove that given an ergodic (globally) invariant measure with a given marginal, for almost every fiber the corresponding conditional measure has dimension equal to the ratio of the relative metric entropy and the Lyapunov exponent. Finally we show that there is exactly one invariant measure whose conditional measures are of full dimension. It is the canonical Gibbs state.

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 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.

Measures of maximal relative entropy with full support

2010 ◽  
Vol 31 (6) ◽  
pp. 1889-1899 ◽  
Author(s):  
JISANG YOO
Keyword(s):  
Probability Measure ◽  
Finite Type ◽  
Relative Entropy ◽  
Invariant Probability Measure ◽  
Shift Of Finite Type ◽  
Shift Spaces ◽  
Full Support ◽  
Shift Space ◽  
Sofic Shifts ◽  
Factor Map

AbstractLet π be a factor map from an irreducible shift of finite type X to a shift space Y. Let ν be an invariant probability measure on Y with full support. We show that every measure on X of maximal relative entropy over ν is fully supported. As a result, given any invariant probability measure ν on Y with full support, there is an invariant probability measure μ on X with full support that maps to ν under π. If ν is ergodic, μ can be chosen to be ergodic. These results can be generalized to the case of sofic shifts. We demonstrate that the results do not extend to general shift spaces by providing counterexamples.

scholarly journals Bounded-to-1 factors of an aperiodic shift of finite type are 1-to-1 almost everywhere factors also

1990 ◽  
Vol 10 (4) ◽  
pp. 615-625 ◽  
Author(s):  
Jonathan Ashley
Keyword(s):  
Finite Type ◽  
Shift Of Finite Type ◽  
Almost Everywhere ◽  
Factor Map

AbstractWe show that if π: ΣG → ΣH is a bounded-to-1 factor map from an irreducible shift of finite type ΣG with period pG to a shift of finite type ΣH with period pH, then there is a factor map that is (pG/pH)-to-1 almost everywhere. Moreover, if π is right closing, then may be taken to be right closing also.

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 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.

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.

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