Symbolic dynamics on amenable groups: the entropy of generic shifts

Topological Entropy of Sets of Generic Points for Actions of Amenable Groups

Let G be a countable discrete amenable group which acts continuously on a compact metric space X and let μ be an ergodic G−invariant Borel probability measure on X. For a fixed tempered Følner sequence {Fn} inG with lim n→+∞ |Fn| logn =∞, we prove the following variational principle: h(Gμ, {Fn}) = hμ(X,G), where Gμ is the set of generic points for μ with respect to {Fn} and h(Gμ, {Fn}) is the B...

Symbolic Dynamics on Free Groups

We study nearest-neighbor shifts of finite type (NNSOFT) on a free group G. We determine when a NNSOFT on G admits a periodic coloring and give an example of a NNSOFT that does not allow a periodic coloring. Then, we find an expression for the entropy of the golden mean shift on G. In doing so, we study a new generalization of Fibonacci numbers and analyze their asymptotics with a one-dimension...

Dual Entropy in Discrete Groups with Amenable Actions

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) = hTop(γ̂) where hTop(γ̂) denotes the topological entropy of the induced automorphism γ̂ of the (compact, abelian) dual...

Ergodic Theorems on Amenable Groups

Amenable Groups

Throughout we let Γ be a discrete group. For f : Γ → C and each s ∈ Γ we define the left translation action by (s.f)(t) = f(s−1t). Definition 1.1. A group Γ is amenable is there exists a state μ on l∞(Γ) which is invariant under the left translation action: for all s ∈ Γ and f ∈ l∞(Γ), μ(s.f) = μ(f). Example 1.2. Finite groups are amenable: take the state which sends χ{s} to 1 |Γ| for each s ∈ ...