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 (18)

• Ising models on locally tree-like graphs
• Entropy and the variational principle for actions of sofic groups
• Orbit inequivalent actions of non-amenable groups
• Fuglede-Kadison Determinants and Sofic Entropy
• A sofic group away from amenable groups
• Amenability of groups is characterized by Myhill's Theorem
• Weak containment and Rokhlin entropy
• Free Groups in Lattices
• Orbit equivalence rigidity for ergodic actions of the mapping class group
• Polish Models and Sofic Entropy
• Mixing and Spectral Gap Relative to Pinsker Factors for Sofic Groups
• Sofic groups: graph products and graphs of groups
• Positive sofic entropy implies finite stabilizer
• Relative entropy and the Pinsker product formula for sofic groups
• Sofic Entropy of Gaussian Actions
• On sofic approximations of Property (T) groups
• The Wreath Product of Two Sofic Groups is Sofic
• Subshifts of finite type which have completely positive entropy

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• 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 ${\mathbb{F}}_2\times {\mathbb{F}}_2$