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Strong ergodicity phenomena for Bernoulli shifts of bounded algebraic dimension

Aristotelis Panagiotopoulos, Assaf Shani

Abstract

The algebraic dimension of a Polish permutation group $Q\leq \mathrm{Sym}(\mathbb{N})$ is the smallest $n\inω$, so that for all $A\subseteq \mathbb{N}$ of size $n+1$, the orbit of every $a\in A$ under the pointwise stabilizer of $A\setminus\{a\}$ is finite. We study the Bernoulli shift $P\curvearrowright \mathbb{R}^{\mathbb{N}}$ for various Polish permutation groups $P$ and we provide criteria under which the $P$-shift is generically ergodic relative to the injective part of the $Q$-shift, when $Q$ has algebraic dimension $\leq n$. We use this to show that the sequence of pairwise $*$-reduction-incomparable equivalence relations defined in [KP21] is a strictly increasing sequence in the Borel reduction hierarchy. We also use our main theorem to exhibit an equivalence relation of pinned cardinal $\aleph_1^{+}$ which strongly resembles the equivalence relation of pinned cardinal $\aleph_1^{+}$ from [Zap11], but which does not Borel reduce to the latter. It remains open whether they are actually incomparable under Borel reductions. Our proofs rely on the study of symmetric models whose symmetries come from the group $Q$. We show that when $Q$ is "locally finite" -- e.g. when $Q=\mathrm{Aut}(\mathcal{M})$, where $\mathcal{M}$ is a locally finite countable structure with no algebraicity -- the corresponding symmetric model admits a theory of supports which is analogous to that in the basic Cohen model.

Strong ergodicity phenomena for Bernoulli shifts of bounded algebraic dimension

Abstract

The algebraic dimension of a Polish permutation group is the smallest , so that for all of size , the orbit of every under the pointwise stabilizer of is finite. We study the Bernoulli shift for various Polish permutation groups and we provide criteria under which the -shift is generically ergodic relative to the injective part of the -shift, when has algebraic dimension . We use this to show that the sequence of pairwise -reduction-incomparable equivalence relations defined in [KP21] is a strictly increasing sequence in the Borel reduction hierarchy. We also use our main theorem to exhibit an equivalence relation of pinned cardinal which strongly resembles the equivalence relation of pinned cardinal from [Zap11], but which does not Borel reduce to the latter. It remains open whether they are actually incomparable under Borel reductions. Our proofs rely on the study of symmetric models whose symmetries come from the group . We show that when is "locally finite" -- e.g. when , where is a locally finite countable structure with no algebraicity -- the corresponding symmetric model admits a theory of supports which is analogous to that in the basic Cohen model.

Paper Structure

This paper contains 14 sections, 13 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Definitions and preliminaries
  3. Borel reductions and strong ergodicity
  4. Bernoulli shifts
  5. Automorphism groups and ultrahomogeneous structures
  6. The model-theoretic algebraic closure
  7. Examples
  8. Labelled logic actions
  9. A symmetric model argument
  10. Symmetric models
  11. Absolute classification
  12. Proofs of the main results
  13. A strictly $\leq_B$-increasing sequence
  14. Unpinned equivalence relations

Key Result

Theorem \oldthetheorem

Let $P,Q\leq \mathrm{Sym}(\mathbb{N})$ be Polish permutation groups and let $n\in\omega$. If the algebraic dimension of $Q$ is at most $n$ and $P$ is locally finite and $(n+1)$-free, then $E_{\mathrm{inj}}(P)$ is generically $E_{\mathrm{inj}}(Q)$-ergodic. As a consequence, we have that:

Theorems & Definitions (42)

  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • ...and 32 more