Dual entropy in discrete groups with amenable actions

Abstract

Let G be a discrete group which admits an amenable action on a compact space and γ ∈ Aut(G) be an automorphism. We define a notion of entropy for γ and denote the invariant by ha(γ). This notion is dual to classical topological entropy in the sense that if G is abelian then ha(γ) = hT op(γ) where hT op(γ) denotes the topological entropy of the induced automorphismγ of the (compact, abelian) dual groupĜ.ha(·) enjoys a number of basic properties which are useful for calculations. For example, it decreases in invariant subgroups and certain quotients. These basic properties are used to compute the dual entropy of an arbitrary automorphism of a crystallographic group.

Non-commutative entropy computations for continuous fields and cross-products

“…Then we can reinterpret proposition 3.3 of [7]. This proposition actually shows that if one chooses a finite set Ω of elements of Z 2 (and sees them as monomials in the two generators U and V ) then sup θ rcp( , It remains to compute the symbolic entropy of the automorphism.…”

“…In a private communication , N.P. Brown mentionned that a computation of the "dual entropy" (see [7]) of the same automorphism of the field of non-commutative tori, but this time seen as an automorphism of the C * -algebra of the Heisenberg group is possible and that one gets the same upper bound.…”

Entropy for automorphisms of free groups

“…Based on this fact, Brown-Germain [4] defined the entropy ha(α) for an automorphism α of an exact discrete group.…”

“…In this section, we summarize notations, terminologies and basic facts on the entropy ha(α) in [4] for an automorphism α of an exact discrete group G.…”

“…Proof. Let α be an automorphism of an exact discrete group G. It always holds that ha(α) ≥ ht(α) by [4,Proposition 3.3]. If β is an automorphism of an exact C * -algebra, then ht(β) ≥ h φ (β) by [8,Proposition 9], and also for the modified dynamical entropy ht φ (β) of ht(β), we have that ht φ (β) ≤ ht(β) by [5] (cf.…”