A Brief Introduction to Sofic Entropy Theory

Bowen, Lewis

doi:10.1142/9789813272880_0120

Cited by 12 publications

(12 citation statements)

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“…Apparently the definition of entropy considered here and in [14,15] is not the same as sofic entropy (see [6,9]), since the latter is a conjugacy invariant while the entropy considered here can increase under higher block codes.…”

Section: Introductionmentioning

confidence: 92%

“…In recent years increasing attention has been paid to the calculation of entropy for systems in which the "time" is not Z nor R, but perhaps Z d for some d ≥ 2, or an arbitrary amenable group, or even a free or arbitrary countable group. We will not attempt to review the extensive and rapidly developing literature here (nor the connections with information theory, statistical mechanics, and other areas), referring only to [6,8,9] for background and references.…”

Section: Introductionmentioning

confidence: 99%

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Entropy on regular trees

Petersen¹,

Salama²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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We show that the limit in our definition of tree shift topological entropy is actually the infimum, as is the case for both the topological and measure-theoretic entropies in the classical situation when the time parameter is Z. As a consequence, tree shift entropy becomes somewhat easier to work with. For example, the statement that the topological entropy of a tree shift defined by a one-dimensional subshift dominates the topological entropy of the latter can now be extended from shifts of finite type to arbitrary subshifts. Adapting to trees the strip method already used to approximate the hard square constant on Z 2 , we show that the entropy of the hard square tree shift on the regular k-tree increases with k, in contrast to the case of Z k . We prove that the strip entropy approximations increase strictly to the entropy of the golden mean tree shift for k = 2, . . . , 8 and propose that this holds for all k ≥ 2. We study the dynamics of the map of the simplex that advances the vector of ratios of symbol counts as the width of the approximating strip is increased, providing a fairly complete description for the golden mean subshift on the k-tree for all k. This map provides an efficient numerical method for approximating the entropies of tree shifts defined by nearest neighbor restrictions. Finally, we show that counting configurations over certain other patterns besides the natural finite subtrees yields the same value of entropy for tree SFT's.

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Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Entropy on regular trees

Petersen¹,

Salama²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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“…The skew product approach for non-commutative transformations leads to a not well-investigated notion of entropy [ 155 , 156 ]. It would be interesting to compare this approach with the approach of L. Bowen for the definition of entropy of free group action [ 157 , 158 ].…”

Section: Random Model and Concluding Remarksmentioning

confidence: 99%

Integrable and Chaotic Systems Associated with Fractal Groups

Grigorchuk

Samarakoon

2021

Entropy

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Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor’s question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schrödinger operators. One important development is the relation of fractal groups to multi-dimensional dynamics, the theory of joint spectrum of pencil of operators, and the spectral theory of Laplace operator on graphs. This paper gives a quick access to these topics, provides calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains a discussion of the dichotomy “integrable-chaotic” in the considered model, and suggests a possible probabilistic approach to studying the discussed problems.

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“…These invariants generalize classical invariants of Z-actions. For an introduction to sofic entropy, see [Bow18]. This motivates the problem: generalize these invariants to actions by locally compact groups.…”

Section: Group Ringsmentioning

confidence: 99%