On the clustering conjecture for Bernoulli factors of 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...

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

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.

On a conjecture about numerators of the Bernoulli numbers

In this paper we disprove a conjecture about numerators of divided Bernoulli numbers Bn/n and Bn/n(n − 1) which was suggested by Roland Bacher. We give some counterexamples. Subsequently we extend the results to the general case.