Entropy and the variational principle for actions of sofic groups

By David Kerr, Hanfeng Li. 2010

149 citations recorded by the source corpus.

Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy. By applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants, and we establish the variational principle in this context. In the case of residually finite groups we use the variational principle to compute the topological entropy of principal algebraic actions whose defining group ring element is invertible in the full group C∗-algebra.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a600af02bfd3f9ed1b03be64d2942730cc57213d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Compact Group Automorphisms, Addition Formulas and Fuglede-Kadison Determinants (2010, background): “Recently, entropy has been defined for continuous actions of a countable sofic group on compact metrizable spaces and measure-preserving actions of a countable sofic group on standard probability measure spaces, with respect to a sofic approximation sequence of the sofic group [4, 34].”
Soficity, amenability, and dynamical entropy (2010, methodology): “The goal of this paper is to prove that, when the acting group is amenable, the sofic measure and topological entropies from [9] both coincide with their classical counterparts, independently of the sofic approximation sequence.”
Homoclinic groups, IE groups, and expansive algebraic actions (2011, unknown): “D S] 2 J an 2 01”
Combinatorial Independence and Sofic Entropy (2012, methodology): “The authors developed in [26] a more general operator-algebraic approach to sofic entropy that enables one to remove the generator hypothesis (see also [23] for a formulation in terms of finite partitions).”
Variational principles for topological entropies of subsets (2010, background): “Recently, Kerr and Li obtained a variational principle of topological entropy and measure-theoretic entropy for actions of sofic group [19,20].”
Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture (2013, background): “Another proof was obtained Kerr and Li in [75] as an application of the theory of entropy for actions of sofic groups developed by Bowen, Kerr, and Li (see [7], [75], and [76]).”
ENTROPY, DETERMINANTS, AND L 2 -TORSION (2012, background): “Recently, entropy theory has been extended to actions of countable sofic groups [5], [36], which include all countable amenable groups and countable residually finite groups.”
Sofic measure entropy via finite partitions (2011, methodology): “This leads to a completely general notion of sofic measure entropy, as well as a topological counterpart and a variational principle relating the two [5].”