scholarly journals The Chaotic Properties of Increasing Gap Shifts

2019 ◽  
Vol 2019 ◽  
pp. 1-5
Author(s):  
Nor Syahmina Kamarudin ◽  
Malouh Baloush ◽  
Syahida Che Dzul-Kifli
Keyword(s):  
Finite Type ◽  
Shift Of Finite Type ◽  
Infinite Type ◽  
Topologically Mixing

It is well known that locally everywhere onto, totally transitive, and topologically mixing are equivalent on shift of finite type. It turns out that this relation does not hold true on shift of infinite type. We introduce the increasing gap shift and determine its chaotic properties. The increasing gap shift and the sigma star shift serve as counterexamples to show the relation between the three chaos notions on shift of infinite type.

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.

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.

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.

scholarly journals -action Induced by Shift Map on 1-Step Shift of Finite Type over Two Symbols and k-type Transitive

2020 ◽  
Vol 8 (5) ◽  
pp. 535-541
Author(s):  
Nor Syahmina Kamarudin ◽  
Syahida Che Dzul-Kifli
Keyword(s):  
Finite Type ◽  
Shift Of Finite Type ◽  
Shift Map

scholarly journals Rigid hypersurfaces and infinitesimal CR automorphisms

Filomat ◽  
2017 ◽  
Vol 31 (1) ◽  
pp. 77-84
Author(s):  
Atsushi Hayashimoto
Keyword(s):  
Finite Type ◽  
Open Problems ◽  
Infinite Type ◽  
Infinitesimal Cr Automorphisms

We study a survey on the relations between rigid hypersurfaces and infinitesimal CR automorphisms. After reviewing the case of hypersurfaces of finite type, we study the case of hypersurfaces of infinite type. Some open problems are posed in the last section.

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