Bernoulli shifts induce Bernoulli shifts

Weak isomorphisms between Bernoulli shifts

In this note, we prove that if G is a countable group that contains a nonabelian free subgroup then every pair of nontrivial Bernoulli shifts over G are weakly isomorphic.

Brooks’ theorem for Bernoulli shifts

If Γ is an infinite group with finite symmetric generating set S, we consider the graph G(Γ, S) on [0, 1]Γ by relating two distinct points if an element of s sends one to the other via the shift action. We show that, aside from the cases Γ = Z and Γ = (Z/2Z) ∗ (Z/2Z), G(Γ, S) satisfies a measure-theoretic version of Brooks’ theorem: there is a G(Γ, S)-invariant conull Borel set B ⊆ [0, 1]Γ and ...

Institute for Mathematical Physics Resolving Markov Chains onto Bernoulli Shifts Resolving Markov Chains onto Bernoulli Shifts

Z Group Shifts and Bernoulli Factors

In this paper, a group shift is an expansive action of Zd on a compact metrizable zero dimensional group by continuous automorphisms. All group shifts factor topologically onto equal-entropy Bernoulli shifts; abelian group shifts factor by continuous group homomorphisms onto canonical equalentropy Bernoulli group shifts; and completely positive entropy abelian group shifts are weakly algebraica...

Positive Entropy Actions of Countable Groups Factor onto Bernoulli Shifts

We prove that if a free ergodic action of a countably infinite group has positive Rokhlin entropy (or, less generally, positive sofic entropy) then it factors onto all Bernoulli shifts of lesser or equal entropy. This extends to all countably infinite groups the well-known Sinai factor theorem from classical entropy theory. We also use our methods to deduce spectral properties of positive entro...