scholarly journals Positive Sofic Entropy Implies Finite Stabilizer

Entropy ◽  
2016 ◽  
Vol 18 (7) ◽  
pp. 263 ◽  
Author(s):  
Tom Meyerovitch
Keyword(s):  
Sofic Entropy

scholarly journals Uniform mixing and completely positive sofic entropy

2019 ◽  
Vol 138 (2) ◽  
pp. 597-612
Author(s):  
Tim Austin ◽  
Peter Burton
Keyword(s):  
Completely Positive ◽  
Sofic Entropy

scholarly journals ADDITIVITY PROPERTIES OF SOFIC ENTROPY AND MEASURES ON MODEL SPACES

2016 ◽  
Vol 4 ◽  
Author(s):  
TIM AUSTIN
Keyword(s):  
Lower Bound ◽  
Probability Distributions ◽  
Cartesian Product ◽  
Sufficient Conditions ◽  
Entropy Theory ◽  
Reverse Inequality ◽  
Cartesian Products ◽  
Model Spaces ◽  
Sofic Entropy ◽  
Sofic Groups

Sofic entropy is an invariant for probability-preserving actions of sofic groups. It was introduced a few years ago by Lewis Bowen, and shown to extend the classical Kolmogorov–Sinai entropy from the setting of amenable groups. Some parts of Kolmogorov–Sinai entropy theory generalize to sofic entropy, but in other respects this new invariant behaves less regularly. This paper explores conditions under which sofic entropy is additive for Cartesian products of systems. It is always subadditive, but the reverse inequality can fail. We define a new entropy notion in terms of probability distributions on the spaces of good models of an action. Using this, we prove a general lower bound for the sofic entropy of a Cartesian product in terms of separate quantities for the two factor systems involved. We also prove that this lower bound is optimal in a certain sense, and use it to derive some sufficient conditions for the strict additivity of sofic entropy itself. Various other properties of this new entropy notion are also developed.

scholarly journals Sofic entropy and amenable groups

2011 ◽  
Vol 32 (2) ◽  
pp. 427-466 ◽  
Author(s):  
LEWIS BOWEN
Keyword(s):  
Group Action ◽  
Amenable Group ◽  
Group Actions ◽  
Amenable Groups ◽  
Orbit Equivalent ◽  
Sofic Entropy ◽  
Classical Entropy

AbstractIn previous work, the author introduced a measure-conjugacy invariant for sofic group actions called sofic entropy. Here, it is proven that the sofic entropy of an amenable group action equals its classical entropy. The proof uses a new measure-conjugacy invariant called upper-sofic entropy and a theorem of Rudolph and Weiss for the entropy of orbit-equivalent actions relative to the orbit changeσ-algebra.

scholarly journals Combinatorial Independence and Sofic Entropy

2013 ◽  
Vol 1 (2) ◽  
pp. 213-257 ◽  
Author(s):  
David Kerr ◽  
Hanfeng Li
Keyword(s):  
Sofic Entropy

scholarly journals Sofic entropy of Gaussian actions

2016 ◽  
Vol 37 (7) ◽  
pp. 2187-2222 ◽  
Author(s):  
BEN HAYES
Keyword(s):  
Probability Measure ◽  
Fourier Analysis ◽  
Harmonic Analysis ◽  
Discrete Group ◽  
Orthogonal Representation ◽  
Countable Discrete Group ◽  
Abelian Case ◽  
Sofic Entropy ◽  
Model Formalism ◽  
Invent Math

Associated to any orthogonal representation of a countable discrete group, is a probability measure-preserving action called the Gaussian action. Using the Polish model formalism we developed before, we compute the entropy (in the sense of Bowen [J. Amer. Math. Soc.23(2010) 217–245], Kerr and Li [Invent. Math.186(2011) 501–558]) of Gaussian actions when the group is sofic. Computation of entropy for Gaussian actions has only been done when the acting group is abelian and thus our results are new, even in the amenable case. Fundamental to our approach are methods of non-commutative harmonic analysis and$C^{\ast }$-algebras which replace the Fourier analysis used in the abelian case.

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 Examples in the entropy theory of countable group actions

2019 ◽  
Vol 40 (10) ◽  
pp. 2593-2680 ◽  
Author(s):  
LEWIS BOWEN
Keyword(s):  
Classical Theory ◽  
Group Actions ◽  
Countable Group ◽  
Classification Theory ◽  
Entropy Theory ◽  
Survey Article ◽  
Sofic Entropy ◽  
Factor Maps ◽  
Measure Preserving Actions ◽  
Classical Entropy

Kolmogorov–Sinai entropy is an invariant of measure-preserving actions of the group of integers that is central to classification theory. There are two recently developed invariants, sofic entropy and Rokhlin entropy, that generalize classical entropy to actions of countable groups. These new theories have counterintuitive properties such as factor maps that increase entropy. This survey article focusses on examples, many of which have not appeared before, that highlight the differences and similarities with classical theory.

Sign in / Sign up

Export Citation Format

Share Document