Properly ergodic structures

TL;DR

This paper studies ergodic invariant probability measures on countable structures, characterizes theories admitting such measures, and explores the diversity of models in the context of ergodic theory and logic.

Contribution

It characterizes theories with properly ergodic structures and analyzes the model-theoretic properties of their almost-sure theories.

Findings

Almost-sure $F$-theory of a properly ergodic structure has continuum-many models.

Full almost-sure $ ext{L}_{oldsymbol{ ext{ω}_1, ext{ω}}}$-theory has no models.

Existence of properly ergodic structures concentrated on models of a theory implies continuum-many such structures.

Abstract

We consider ergodic $\mathrm{Sym}(\mathbb{N})$-invariant probability measures on the space of $L$-structures with domain $\mathbb{N}$ (for $L$ a countable relational language), and call such a measure a properly ergodic structure when no isomorphism class of structures is assigned measure $1$. We characterize those theories in countable fragments of $\mathcal{L}_{\omega_1, \omega}$ for which there is a properly ergodic structure concentrated on the models of the theory. We show that for a countable fragment $F$ of $\mathcal{L}_{\omega_1, \omega}$ the almost-sure $F$-theory of a properly ergodic structure has continuum-many models (an analogue of Vaught's Conjecture in this context), but its full almost-sure $\mathcal{L}_{\omega_1, \omega}$-theory has no models. We also show that, for an $F$-theory $T$, if there is some properly ergodic structure that concentrates on the class of models of $T$, then there are continuum-many such properly ergodic structures.