Conditional entropy and fiber entropy for amenable group actions

Abstract

In this paper, we introduce the notions of topological conditional entropy and fiber entropy for a given factor map between two amenable group actions, and prove three variational principles for conditional entropy and fiber entropy. Moreover, as an application of our variational principle we prove that the countable-to-one extension and the distal extension have zero conditional topological entropy. In particular, the topological entropy of a distal system is zero.

Relative Topological Entropy for Actions of Non-discrete Groups on Compact Spaces in the Context of Cut and Project Schemes

“…Remark 5.3. Theorem 5.2 shows that the relative topological entropy considered in [Yan15] for actions of countable discrete groups on compact metric spaces is equivalent to our definition. In order to see this combine [Yan15, Lemma 2.4.]…”

“…Let π be an action of a amenable group G, containing a countable uniform lattice Λ, on a compact Hausdorff space X and let ϕ be a factor of π via p : X → Y . Then In Remark 5.3 it is presented, that our definition of relative topological entropy is equivalent to the definition given in [Yan15]. As [Yan15, Lemma 5.4] is also valid in the context of compact Hausdorff spaces the proof given in [Yan15, Theorem 5.1] easily generalizes to actions on compact Hausdorff spaces.…”

“…For A ∈ B X and ν ∈ M π let E ν,p (χ A ) be the conditional expectation of the characteristic function χ A of A with respect to p −1 (B X ). For α ∈ P X we define Using the variational principle, shown in [Yan15] for discrete countable groups, we obtain the following. Theorem 6.5.…”

“…Different authors considered topological entropy for actions of countable discrete amenable groups. See for example [OW87,HYZ10,Yan15,ZC16,Zho16]. These definitions are independent from the choice of the chosen Van Hove sequence.…”

“…ν,p (α) := − A∈α X E ν,p (χ A ) log(E ν,p (χ A ))dν.As presented in[Yan15] the Ornstein-Weiss lemma can be applied to F (Λ) ∋ F → H ν,p (α F ) for every α ∈ P X to obtain thatE ν (α|π p → ϕ) := lim i∈I H ν,p (α Fi ) |F i |exists and that is independent of the choice of the Van Hove net (F i ) i∈I in Λ. The relative measure theoretical entropy of π under the condition ϕ is given byE ν (π p → ϕ) := sup α∈PX E ν (α|π p → ϕ).The following can be found in [Yan15, Theorem 3.1].Proposition 6.4.…”