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29
Weakly almost periodic functions, model theoretic stability, and minimality of topological groups
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INVARIANT MEASURES CONCENTRATED ON COUNTABLE STRUCTURES
"... Abstract. Let L be a countable language. We say that a countable infinite Lstructure M admits an invariant measure when there is a probability measure on the space of Lstructures with the same underlying set as M that is invariant under permutations of that set, and that assigns measure one to the ..."
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Abstract. Let L be a countable language. We say that a countable infinite Lstructure M admits an invariant measure when there is a probability measure on the space of Lstructures with the same underlying set as M that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of M. We show that M admits an invariant measure if and only if it has trivial definable closure, i.e., the pointwise stabilizer in Aut(M) of an arbitrary finite tuple of M fixes no additional points. When M is a Fraïssé limit in a relational language, this amounts to requiring that the age of M have strong amalgamation. Our results give rise to new instances of structures that admit
The group of homeomorphisms of the Cantor set has ample generics, preprint
"... Abstract. We show that the group of homeomorphisms of the Cantor set H(2N) has ample generics, that is, for every m the diagonal conjugacy action g · (h1, h2,..., hm) = (gh1g −1, gh2g−1,..., ghmg−1) of H(2N) on H(2N)m has a comeager orbit. This answers a question of Kechris and Rosendal. We show tha ..."
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Abstract. We show that the group of homeomorphisms of the Cantor set H(2N) has ample generics, that is, for every m the diagonal conjugacy action g · (h1, h2,..., hm) = (gh1g −1, gh2g−1,..., ghmg−1) of H(2N) on H(2N)m has a comeager orbit. This answers a question of Kechris and Rosendal. We show that a generic tuple in H(2N)m can be taken to be the limit of a certain projective Fräısse ́ family. We also give an example of a projective Fräısse ́ family, which has a simpler description than the one considered in the general case, and such that its limit is a homeomorphism of the Cantor set that has a comeager conjugacy class. 1.
Homogeneous 1based structures and interpretability in random structures
, 2014
"... Let V be a finite relational vocabulary in which no symbol has arity greater than 2. LetM be countable Vstructure which is homogeneous, simple and 1based. The first main result says that ifM is, in addition, primitive, then it is strongly interpretable in a random structure. The second main resul ..."
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Let V be a finite relational vocabulary in which no symbol has arity greater than 2. LetM be countable Vstructure which is homogeneous, simple and 1based. The first main result says that ifM is, in addition, primitive, then it is strongly interpretable in a random structure. The second main result, which generalizes the first, implies (without the assumption on primitivity) that if M is “coordinatized ” by a set with SUrank 1 and there is no definable (without parameters) nontrivial equivalence relation on M with only finite classes, then M is strongly interpretable in a random structure.
Reachability analysis of firstorder definable pushdown systems
 University of Warsaw
, 2011
"... We study pushdown systems where control states, stack alphabet, and transition relation, instead of being finite, are firstorder definable in a fixed countablyinfinite structure. We show that the reachability analysis can be addressed with the wellknown saturation technique for the wide class of ..."
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We study pushdown systems where control states, stack alphabet, and transition relation, instead of being finite, are firstorder definable in a fixed countablyinfinite structure. We show that the reachability analysis can be addressed with the wellknown saturation technique for the wide class of oligomorphic structures. Moreover, for the more restrictive homogeneous structures, we are able to give concrete complexity upper bounds. We show ample applicability of our technique by presenting several concrete examples of homogeneous structures, subsuming, with optimal complexity, known results from the literature. We show that infinitely many such examples of homogeneous structures can be obtained with the classical wreath product construction.