2021/08/06 by Tomasz Downarowicz, Downarowicz, Tomasz, Piotr Oprocha +5 · 1 citation
Computer Science · Mathematics · #37A15 #37A35 (Primary) #43A07 (Secondary) #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2108.03211
openalex publication_date 2021/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The paper offers a thorough study of multiorders and their applications to measure-preserving actions of countable amenable groups. By a~\em multiorder on a~countable group we mean any probability measure ν on the collection O of linear orders of type \mathbb Z on G, invariant under the natural action of G on such orders. Every free measure-preserving G-action (X,μ,G) has a~multiorder (O,ν,G) as a factor and has the same orbits as the \mathbb Z-action (X,μ,S), where S is the successor map determined by the multiorder factor. Moreover, the sub-sigma-algebra Σ_O associated with the multiorder factor is invariant under S, which makes the corresponding \mathbb Z-action (O,ν, S) a factor of (X,μ,S). We prove that the entropy of any G-process generated by a finite partition of X, conditional with respect to Σ_O, is preserved by the orbit equivalence with (X,μ,S). Furthermore, this entropy can be computed in terms of the so-called random past, by a formula analogous to h(μ,T,\mathcal P)=H(μ,\mathcal P|P-) known for \mathbb Z-actions. The above fact is then applied to prove a variant of a result by Rudolph and Weiss. The original theorem states that orbit equivalence between free actions of countable amenable groups preserves conditional entropy with respect to a~sub-sigma-algebra Σ, as soon as the ``orbit change'' is measurable with respect to Σ. In our variant, we replace the measurability assumption by a~simpler one: Σ should be invariant under both actions and the actions on the resulting factor should be free. In conclusion we provide a characterization of the Pinsker sigma-algebra of any G-process in terms of an appropriately defined remote past arising from a multiorder.