Two-Dimensional Sofic Systems and Shift of Finite Type using Allowable Block

2011 ◽  
Vol 4 (3) ◽  
pp. 109-112
Author(s):  
Abdulkafi A. Al-Refaei
Keyword(s):  
Finite Type ◽  
Two Dimensional ◽  
Shift Of Finite Type

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.

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.

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.

Zeta functions for two-dimensional shifts of finite type

2012 ◽  
Vol 221 (1037) ◽  
pp. 1 ◽  
Author(s):  
Jung-Chao Ban ◽  
Wen-Guei Hu ◽  
Song-Sun Lin ◽  
Yin-Heng Lin
Keyword(s):  
Finite Type ◽  
Zeta Functions ◽  
Two Dimensional

scholarly journals Hartman’s theorem for hyperbolic sets

1977 ◽  
Vol 67 ◽  
pp. 41-52 ◽  
Author(s):  
Masahiro Kurata
Keyword(s):  
Fixed Point ◽  
Finite Type ◽  
Conjugacy Problem ◽  
Topological Conjugacy ◽  
Shift Of Finite Type ◽  
Linear Map ◽  
Hyperbolic Set ◽  
Slight Extension ◽  
Hyperbolic Fixed Point ◽  
Hyperbolic Sets

Hartman proved that a diffeomorphism is topologically conjugate to a linear map on a neighbourhood of a hyperbolic fixed point ([3]). In this paper we study the topological conjugacy problem of a diffeomorphism on a neighbourhood of a hyperbolic set, and prove that for any hyperbolic set there is an arbitrarily slight extension to which a sub-shift of finite type is semi-conjugate.

The action of inert finite-order automorphisms on finite subsystems of the shift

1991 ◽  
Vol 11 (3) ◽  
pp. 413-425 ◽  
Author(s):  
Mike Boyle ◽  
Ulf-Rainer Fiebig
Keyword(s):  
Finite Type ◽  
Finite Order ◽  
Shift Of Finite Type ◽  
Group Of Automorphisms ◽  
Dimension Representation

AbstractLet (X, S) be a shift of finite type. Let G be the group of automorphisms of (X, S) which are compositions of elements of finite order in the kernel of the dimension representation. We characterize the action of G on finite subsystems of (X, S).

Tilings, fundamental cocycles and fundamental groups of symbolic ${\Bbb Z}^{d}$-actions

1998 ◽  
Vol 18 (6) ◽  
pp. 1473-1525 ◽  
Author(s):  
KLAUS SCHMIDT
Keyword(s):  
Finite Type ◽  
Homomorphic Image ◽  
Discrete Group ◽  
Fundamental Groups ◽  
Two Dimensional ◽  
Shift Spaces ◽  
Shift Space ◽  
Topologically Mixing

We prove that certain topologically mixing two-dimensional shifts of finite type have a ‘fundamental’ $1$-cocycle with the property that every continuous $1$-cocycle on the shift space with values in a discrete group is continuously cohomologous to a homomorphic image of the fundamental cocycle. These fundamental cocycles are closely connected with representations of the shift space by Wang tilings and the tiling groups of Conway, Lagarias and Thurston, and they determine the projective fundamental groups of the shift spaces introduced by Geller and Propp.

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