Examples in the entropy theory of countable group actions

Abstract

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.

130 pages. A large number of minor errors corrected from the previous version

References in corpus (4)

• Orbit inequivalent actions of non-amenable groups
• Amenability of groups is characterized by Myhill's Theorem
• Relative entropy and the Pinsker product formula for sofic groups
• On sofic approximations of Property (T) groups

Cited by in corpus (15)

• Entanglement Accessibility Measures for the Quantum Internet
• Markovian properties of continuous group actions: algebraic actions, entropy and the homoclinic group
• Combinatorial independence and naive entropy
• The Lanford-Ruelle theorem for actions of sofic groups
• Random ordering formula for sofic and Rokhlin entropy of Gibbs measures
• New Examples of Bernoulli Algebraic Actions
• Predictability, topological entropy and invariant random orders
• Metastability and maximal-entropy joinings of Gibbs measures on finitely-generated groups
• Kieffer-Pinsker type formulas for Gibbs measures on sofic groups
• Algebraic dynamical systems from LDPC codes satisfy a strong negation of the weak Pinsker property
• Amorphic complexity of group actions with applications to quasicrystals
• Conformal measures and the Dobrushin-Lanford-Ruelle equations
• Bernoullicity of lopsided principal algebraic actions
• Entropy of Toeplitz systems over residually finite groups
• On sofic approximations of ${\mathbb{F}}_2\times {\mathbb{F}}_2$