Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Properly ergodic structures   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.

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The categories of this paper are: plasm-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ex
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The categories of this paper are: q-bio
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The categories of this paper are: solv-int
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The categories of this paper are: alg-geom
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The categories of this paper are: adap-org
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The categories of this paper are: supr-con
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The categories of this paper are: hep-th
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The categories of this paper are: cs
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The categories of this paper are: acc-phys
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The categories of this paper are: math-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: ao-sci