Subshifts of multi-dimensional shifts of finite type

2000 ◽  
Vol 20 (3) ◽  
pp. 859-874 ◽  
Author(s):  
ANTHONY N. QUAS ◽  
PAUL B. TROW
Keyword(s):  
Finite Type ◽  
Infinite Chain ◽  
Positive Entropy ◽  
Shift Of Finite Type ◽  
Subshifts Of Finite Type

We show that every shift of finite type $X$ with positive entropy has proper subshifts of finite type with entropy strictly smaller than the entropy of $X$, but with entropy arbitrarily close to the entropy of $X$. Consequently, $X$ contains an infinite chain of subshifts of finite type which is strictly decreasing in entropy.

scholarly journals Chains, entropy, coding

1986 ◽  
Vol 6 (3) ◽  
pp. 415-448 ◽  
Author(s):  
Karl Petersen
Keyword(s):  
Growth Rate ◽  
Markov Chains ◽  
Random Walks ◽  
Finite Type ◽  
Periodic Orbits ◽  
Entropy Coding ◽  
Positive Entropy ◽  
Loop Analysis ◽  
Countable State ◽  
Subshifts Of Finite Type

AbstractVarious definitions of the entropy for countable-state topological Markov chains are considered. Concrete examples show that these quantities do not coincide in general and can behave badly under nice maps. Certain restricted random walks which arise in a problem in magnetic recording provide interesting examples of chains. Factors of some of these chains have entropy equal to the growth rate of the number of periodic orbits, even though they contain no subshifts of finite type with positive entropy; others are almost sofic – they contain subshifts of finite type with entropy arbitrarily close to their own. Attempting to find the entropies of such subshifts of finite type motivates the method of entropy computation by loop analysis, in which it is not necessary to write down any matrices or evaluate any determinants. A method for variable-length encoding into these systems is proposed, and some of the smaller subshifts of finite type inside these systems are displayed.

scholarly journals A characterization of topologically completely positive entropy for shifts of finite type

2013 ◽  
Vol 34 (6) ◽  
pp. 2054-2065 ◽  
Author(s):  
RONNIE PAVLOV
Keyword(s):  
Dynamical System ◽  
Finite Type ◽  
American Mathematical Society ◽  
Equivalent Condition ◽  
Positive Entropy ◽  
Completely Positive ◽  
Shift Of Finite Type ◽  
Formula Group ◽  
Zero Entropy

AbstractA topological dynamical system was defined by Blanchard [Fully Positive Topological Entropy and Topological Mixing (Symbolic Dynamics and Applications (in honor of R. L. Adler), 135). American Mathematical Society Contemporary Mathematics, Providence, RI, 1992, pp. 95–105] to have topologically completely positive entropy (or TCPE) if its only zero entropy factor is the dynamical system consisting of a single fixed point. For ${ \mathbb{Z} }^{d} $ shifts of finite type, we give a simple condition equivalent to having TCPE. We use our characterization to derive a similar equivalent condition to TCPE for the subclass of ${ \mathbb{Z} }^{d} $ group shifts, which was proved by Lind and Schmidt in the abelian case [Homoclinic points of algebraic ${ \mathbb{Z} }^{d} $-actions. J. Amer. Math. Soc. 12(4) (1999), 953–980] and by Boyle and Schraudner in the general case [${ \mathbb{Z} }^{d} $ group shifts and Bernoulli factors. Ergod. Th. & Dynam. Sys. 28(2) (2008), 367–387]. We also give an example of a ${ \mathbb{Z} }^{2} $ shift of finite type which has TCPE but is not even topologically transitive, and prove a result about block gluing ${ \mathbb{Z} }^{d} $ SFTs motivated by our characterization of TCPE.

scholarly journals Topologically completely positive entropy and zero-dimensional topologically completely positive entropy

2017 ◽  
Vol 38 (5) ◽  
pp. 1894-1922
Author(s):  
RONNIE PAVLOV
Keyword(s):  
Finite Type ◽  
Sufficient Condition ◽  
Positive Entropy ◽  
Two Dimensional ◽  
Completely Positive ◽  
Shift Of Finite Type ◽  
One Dimensional ◽  
Zero Entropy ◽  
The Difference

In a previous paper [Pavlov, A characterization of topologically completely positive entropy for shifts of finite type. Ergod. Th. & Dynam. Sys.34 (2014), 2054–2065], the author gave a characterization for when a $\mathbb{Z}^{d}$-shift of finite type has no non-trivial subshift factors with zero entropy, a property which we here call zero-dimensional topologically completely positive entropy. In this work, we study the difference between this notion and the more classical topologically completely positive entropy of Blanchard. We show that there are one-dimensional subshifts and two-dimensional shifts of finite type which have zero-dimensional topologically completely positive entropy but not topologically completely positive entropy. In addition, we show that strengthening the hypotheses of the main result of Pavlov [A characterization of topologically completely positive entropy for shifts of finite type. Ergod. Th. & Dynam. Sys.34 (2014), 2054–2065] yields a sufficient condition for a $\mathbb{Z}^{d}$-shift of finite type to have topologically completely positive entropy.

On intrinsic ergodicity of factors of subshifts

2015 ◽  
Vol 37 (2) ◽  
pp. 621-645
Author(s):  
KEVIN MCGOFF ◽  
RONNIE PAVLOV
Keyword(s):  
Recent Work ◽  
Finite Type ◽  
Maximal Entropy ◽  
Positive Entropy ◽  
Shift Of Finite Type ◽  
Specification Property ◽  
Unique Factor ◽  
Topologically Mixing ◽  
Mixing Property

It is well known that any $\mathbb{Z}$ subshift with the specification property has the property that every factor is intrinsically ergodic, i.e. every factor has a unique factor of maximal entropy. In recent work, other $\mathbb{Z}$ subshifts have been shown to possess this property as well, including $\unicode[STIX]{x1D6FD}$-shifts and a class of $S$-gap shifts. We give two results that show that the situation for $\mathbb{Z}^{d}$ subshifts with $d>1$ is quite different. First, for any $d>1$, we show that any $\mathbb{Z}^{d}$ subshift possessing a certain mixing property must have a factor with positive entropy which is not intrinsically ergodic. In particular, this shows that for $d>1$, $\mathbb{Z}^{d}$ subshifts with specification cannot have all factors intrinsically ergodic. We also give an example of a $\mathbb{Z}^{2}$ shift of finite type, introduced by Hochman, which is not even topologically mixing, but for which every positive-entropy subshift factor is intrinsically ergodic.

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.

Subshifts of finite type

1976 ◽  
pp. 117-130
Author(s):  
Manfred Denker ◽  
Christian Grillenberger ◽  
Karl Sigmund
Keyword(s):  
Finite Type ◽  
Subshifts Of Finite Type

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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