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.

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.

Entropy of $C^\ast$-dynamical systems defined by bitstreams

1998 ◽  
Vol 18 (4) ◽  
pp. 859-874 ◽  
Author(s):  
V. YA. GOLODETS ◽  
ERLING ST&\Oslash;RMER
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Positive Entropy ◽  
Completely Positive ◽  
Car Algebra

We study automorphisms of the CAR-algebra obtained from binary shifts. We consider cases when the $C^\ast$-dynamical system is asymptotically abelian, is proximally asymptotically abelian, is an entropic $K$-system or has completely positive entropy. The entropy is computed in several cases.

scholarly journals PERIODIC ORBITS OF A DYNAMICAL SYSTEM RELATED TO A KNOT

2011 ◽  
Vol 20 (03) ◽  
pp. 411-426 ◽  
Author(s):  
LILYA LYUBICH
Keyword(s):  
Dynamical System ◽  
Finite Group ◽  
Finite Type ◽  
Periodic Orbits ◽  
Weak Topology ◽  
Commutator Subgroup ◽  
Alexander Polynomial ◽  
Shift Of Finite Type ◽  
A Finite Group

Following [6] we consider a knot group G, its commutator subgroup K = [G, G], a finite group Σ and the space Hom (K, Σ) of all representations ρ : K → Σ, endowed with the weak topology. We choose a meridian x ∈ G of the knot and consider the homeomorphism σx of Hom (K, Σ) onto itself: σxρ(a) = ρ(xax-1) ∀ a ∈ K, ρ ∈ Hom (K, Σ). As proven in [5], the dynamical system ( Hom (K, Σ), σx) is a shift of finite type. In the case when Σ is abelian, Hom (K, Σ) is finite. In this paper we calculate the periods of orbits of ( Hom (K, ℤ/p), σx), where p is prime, in terms of the roots of the Alexander polynomial of the knot. In the case of two-bridge knots we give a complete description of the set of periods.

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 Mean equicontinuity and mean sensitivity

2014 ◽  
Vol 35 (8) ◽  
pp. 2587-2612 ◽  
Author(s):  
JIAN LI ◽  
SIMING TU ◽  
XIANGDONG YE
Keyword(s):  
Dynamical System ◽  
Invariant Measure ◽  
Discrete Spectrum ◽  
Stable System ◽  
Positive Entropy ◽  
Zero Entropy ◽  
Open Question ◽  
Mean Sensitivity ◽  
Spectrum Dichotomy

Answering an open question affirmatively it is shown that every ergodic invariant measure of a mean equicontinuous (i.e. mean-L-stable) system has discrete spectrum. Dichotomy results related to mean equicontinuity and mean sensitivity are obtained when a dynamical system is transitive or minimal. Localizing the notion of mean equicontinuity, notions of almost mean equicontinuity and almost Banach mean equicontinuity are introduced. It turns out that a system with the former property may have positive entropy and meanwhile a system with the latter property must have zero 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.

Entropy theory without a past

2000 ◽  
Vol 20 (5) ◽  
pp. 1355-1370 ◽  
Author(s):  
E. GLASNER ◽  
J.-P. THOUVENOT ◽  
B. WEISS
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Amenable Group ◽  
Entropy Theory ◽  
Positive Entropy ◽  
Product System ◽  
Completely Positive ◽  
Component Systems ◽  
General Setup

This paper treats the Pinsker algebra of a dynamical system in a way which avoids the use of an ordering on the acting group. This enables us to prove some of the classical results about entropy and the Pinsker algebra in the general setup of measure-preserving dynamical systems, where the acting group is a discrete countable amenable group. We prove a basic disjointness theorem which asserts the relative disjointness in the sense of Furstenberg, of $0$-entropy extensions from completely positive entropy (c.p.e.) extensions. This theorem is used to prove several classical results in the general setup. For example, we show that the Pinsker factor of a product system is equal to the product of the Pinsker factors of the component systems. Another application is to obtain a generalization (as well as a simpler proof) of the quasifactor theorem for $0$-entropy systems of Glasner and Weiss.

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.

Sign in / Sign up

Export Citation Format

Share Document