scholarly journals The Spectrum of Completely Positive Entropy Actions of Countable Amenable Groups

2002 ◽  
Vol 196 (1) ◽  
pp. 1-18 ◽  
Author(s):  
A.H. Dooley ◽  
V.Ya. Golodets
Keyword(s):  
Positive Entropy ◽  
Completely Positive ◽  
Amenable Groups

Non-Bernoulli systems with completely positive entropy

2008 ◽  
Vol 28 (1) ◽  
pp. 87-124 ◽  
Author(s):  
A. H. DOOLEY ◽  
V. YA. GOLODETS ◽  
D. J. RUDOLPH ◽  
S. D. SINEL’SHCHIKOV
Keyword(s):  
Infinite Order ◽  
Amenable Group ◽  
Positive Entropy ◽  
Completely Positive ◽  
New Approach ◽  
Amenable Groups ◽  
Uncountable Family ◽  
Cutting And Stacking ◽  
Bernoulli Systems

AbstractA new approach to actions of countable amenable groups with completely positive entropy (cpe), allowing one to answer some basic questions in this field, was recently developed. The question of the existence of cpe actions which are not Bernoulli was raised. In this paper, we prove that every countable amenable groupG, which contains an element of infinite order, has non-Bernoulli cpe actions. In fact we can produce, for any$h \in (0, \infty ]$, an uncountable family of cpe actions of entropyh, which are pairwise automorphically non-isomorphic. These actions are given by a construction which we call co-induction. This construction is related to, but different from the standard induced action. We study the entropic properties of co-induction, proving that ifαGis co-induced from an actionαΓof a subgroup Γ, thenh(αG)=h(αΓ). We also prove that ifαΓis a non-Bernoulli cpe action of Γ, thenαGis also non-Bernoulli and cpe. Hence the problem of finding an uncountable family of pairwise non-isomorphic cpe actions of the same entropy is reduced to one of finding an uncountable family of non-Bernoulli cpe actions of$\mathbb Z$, which pairwise satisfy a property we call ‘uniform somewhat disjointness’. We construct such a family using refinements of the classical cutting and stacking methods.

scholarly journals The Koopman Representation and Positive Rokhlin Entropy

2021 ◽  
Author(s):  
Brandon Seward
Keyword(s):  
Spectral Gap ◽  
Regular Representation ◽  
Positive Entropy ◽  
Completely Positive ◽  
Amenable Groups ◽  
Direct Sum ◽  
Left Regular Representation ◽  
Left Regular ◽  
Sofic Entropy ◽  
Final Consequence

Abstract In this paper, we study connections between positive entropy phenomena and the Koopman representation for actions of general countable groups. Following the line of work initiated by Hayes for sofic entropy, we show in a certain precise manner that all positive entropy must come from portions of the Koopman representation that embed into the left-regular representation. We conclude that for actions having completely positive outer entropy, the Koopman representation must be isomorphic to the countable direct sum of the left-regular representation. This generalizes a theorem of Dooley–Golodets for countable amenable groups. As a final consequence, we observe that actions with completely positive outer entropy must be mixing, and when the group is non-amenable they must be strongly ergodic and have spectral gap.

scholarly journals Garden of Eden and specification

2018 ◽  
Vol 39 (11) ◽  
pp. 3075-3088 ◽  
Author(s):  
HANFENG LI
Keyword(s):  
Positive Entropy ◽  
Completely Positive ◽  
Garden Of Eden ◽  
Amenable Groups ◽  
Equivariant Map ◽  
Specification Property ◽  
Underlying Space ◽  
Algebraic Actions

We establish a Garden of Eden theorem for expansive algebraic actions of amenable groups with the weak specification property, i.e. for any continuous equivariant map $T$ from the underlying space to itself, $T$ is pre-injective if and only if it is surjective. In particular, this applies to all expansive principal algebraic actions of amenable groups and expansive algebraic actions of $\mathbb{Z}^{d}$ with completely positive entropy.

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.

A minimal completely positive entropy non-uniformly positive entropy example

2009 ◽  
Vol 15 (1) ◽  
pp. 87-95 ◽  
Author(s):  
Bailing Song ◽  
Xiangdong Ye
Keyword(s):  
Positive Entropy ◽  
Completely Positive

Bernoulli actions of sofic groups have completely positive entropy

2014 ◽  
Vol 202 (1) ◽  
pp. 461-474 ◽  
Author(s):  
David Kerr
Keyword(s):  
Positive Entropy ◽  
Completely Positive ◽  
Sofic Groups

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.

Co-induction in dynamical systems

2011 ◽  
Vol 32 (3) ◽  
pp. 919-940 ◽  
Author(s):  
ANTHONY H. DOOLEY ◽  
GUOHUA ZHANG
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Amenable Group ◽  
Positive Entropy ◽  
Completely Positive ◽  
Infinite Subgroup

AbstractIf a countable amenable group G contains an infinite subgroup Γ, one may define, from a measurable action of Γ, the so-called co-induced measurable action of G. These actions were defined and studied by Dooley, Golodets, Rudolph and Sinelsh’chikov. In this paper, starting from a topological action of Γ, we define the co-induced topological action of G. We establish a number of properties of this construction, notably, that the G-action has the topological entropy of the Γ-action and has uniformly positive entropy (completely positive entropy, respectively) if and only if the Γ-action has uniformly positive entropy (completely positive entropy, respectively). We also study the Pinsker algebra of the co-induced action.

scholarly journals A hierarchy of topological systems with completely positive entropy

2021 ◽  
Author(s):  
Sebastián Barbieri ◽  
Felipe García-Ramos
Keyword(s):  
Positive Entropy ◽  
Completely Positive

scholarly journals Subshifts of finite type which have completely positive entropy

2011 ◽  
Vol 29 (4) ◽  
pp. 1497-1516 ◽  
Author(s):  
Christopher Hoffman ◽  
Keyword(s):  
Finite Type ◽  
Positive Entropy ◽  
Completely Positive ◽  
Subshifts Of Finite Type
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