Properly Ergodic Structures
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چکیده
We study ergodic Sym(N)-invariant probability measures on the space of L-structures with domain N. We call such measures “ergodic structures”. In particular, we are interested in the properly ergodic case, in which no isomorphism class has measure 1. A Morley–Scott analysis shows that proper ergodicity can always be explained by a splitting of measure over continuum-many types in a countable fragment of Lω1,ω. This implies that the theory of the ergodic structure in any countable fragment of Lω1,ω has continuum-many models up to isomorphism, an analogue of Vaught’s Conjecture in this context. We use the Aldous–Hoover–Kallenberg theorem to show that a structure sampled from a properly ergodic source almost surely satisfies a condition we call “rootedness” on the realizations of the types of measure 0. Finally, we show that a single rooted model of a theory T with trivial definable closure can be used to construct continuum-many distinct properly ergodic structures which almost surely satisfy T . As a consequence, we obtain a characterization of those theories in countable fragments of Lω1,ω which admit properly ergodic models.
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تاریخ انتشار 2017