Separating measurable recurrence from strong recurrence via rigidity sequences

2018/08/16 by John T. Griesmer, Griesmer, John T.
Mathematics · #Limits and Structures in Graph Theory #Graph theory and applications #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1808.05609

Abstract

If G is an abelian group, we say S⊂ G is a set of recurrence if for every probability measure preserving G-system (X,μ,T) and every D⊂ X having μ(D)>0, there is a g∈ S such that μ(D∩ T^gD)>0. We say S is a set of strong recurrence if for every set D having μ(D)>0 there is a c>0 such that μ(D∩ T^gD)>c for infinitely many g∈ S. We call S measure expanding if for all g∈ G, the translate S+g is a set of recurrence. A rigidity sequence for (X,μ,T) is a sequence of elements s_n∈ G satisfying lim_n→∞ μ(D\triangle T^s_nD)=0 for all measurable D⊂ X. For all but countably many countable abelian groups G, we prove that if S is measure expanding, there is a sequence of elements s_n∈ S such that \s_n:n∈ \mathbb N\ is also measure expanding and every translate of (s_n) is a rigidity sequence for some free weak mixing measure preserving G-system. The special case where S=G proves a conjecture of Ackelsberg. As a consequence, we prove that for every countably infinite abelian group G and every measure expanding set S⊂ G there is a subset S'⊂ S such that S' is measure expanding and no translate of S' is a set of strong recurrence.

Related

• Single recurrence in abelian groups
• Rigid-Recurrent Sequences for Actions of Finite Exponent Groups
• Recurrence for stationary group actions
• Coexistence of mixing and rigid behaviors in ergodic theory
• Multiple recurrence and large intersections for abelian group actions