Expansive algebraic actions of discrete residually finite amenable groups and their entropy

By C. Deninger, K. Schmidt. Ergodic Theory and Dynamical Systems, 2006

76 citations recorded by the source corpus.

We prove an entropy formula for certain expansive actions of a countable discrete residually finite group $\Gamma$ by automorphisms of compact abelian groups in terms of Fuglede–Kadison determinants. This extends an earlier result proved by the first author under somewhat more restrictive conditions. The main tools for this generalization are a representation of the $\Gamma$-action by means of a ‘fundamental homoclinic point’ and the description of entropy in terms of the renormalized logarithmic growth rate of the set of $\Gamma_n$-fixed points, where $(\Gamma_n,n\ge1)$ is a decreasing sequence of finite index normal subgroups of $\Gamma$ with trivial intersection.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/043eb159d33ece4868e5326c7a2ebc40161bc128

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Citations in the corpus

Combinatorial Independence and Sofic Entropy (2012, background): “Deninger asked whether this holds for all countable groups [11, Question 26], and affirmative answers were given in [12] for residually finite amenable G and more generally in [9] for amenable G.”
Entropy and the variational principle for actions of sofic groups (2010, unknown): “In the case of amenable acting groups and classical entropy these relationships were developed in [17, 8, 9, 16].”
Homoclinic groups, IE groups, and expansive algebraic actions (2011, background): “We describe first a class of expansive algebraic actions, which are analogues of the expansive principal algebraic actions studied in [17].”
Compact Group Automorphisms, Addition Formulas and Fuglede-Kadison Determinants (2010, background): “After this paper was finished, Douglas Lind informed us that Douglas Lind and Klaus Schmidt have independently obtained results similar to ours in Section 6.”
ENTROPY, DETERMINANTS, AND L 2 -TORSION (2012, background): “Later Deninger-Schmidt [21] showed the equality in the case Γ is amenable and residually finite and f is invertible in 1(Γ).”
Fuglede–Kadison Determinants and Sofic Entropy (2014, background): “Then Deninger-Schmidt in [13] show this equality in the principal case was true when Γ is amenable, residually finite and when f is invertible in l(1)(Γ).”