Quasi-invariant measures concentrating on countable structures
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915802040500224 |
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| author | Conley, Clinton Jahel, Colin Panagiotopoulos, Aristotelis |
| author_facet | Conley, Clinton Jahel, Colin Panagiotopoulos, Aristotelis |
| contents | Countable $\mathcal{L}$-structures $\mathcal{N}$ whose isomorphism class supports a permutation invariant probability measure in the logic action have been characterized by Ackerman-Freer-Patel to be precisely those $\mathcal{N}$ which have no algebraicity. Here we characterize those countable $\mathcal{L}$-structure $\mathcal{N}$ whose isomorphism class supports a quasi-invariant probability measure. These turn out to be precisely those $\mathcal{N}$ which are not "highly algebraic" -- we say that $\mathcal{N}$ is highly algebraic if outside of every finite $F$ there is some $b$ and a tuple $\bar{a}$ disjoint from $b$ so that $b$ has a finite orbit under the pointwise stabilizer of $\bar{a}$ in $\mathrm{Aut}(\mathcal{N})$. As a bi-product of our proof we show that whenever the isomorphism class of $\mathcal{N}$ admits a quasi-invariant measure, then it admits one with continuous Radon--Nikodym cocycles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_07454 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quasi-invariant measures concentrating on countable structures Conley, Clinton Jahel, Colin Panagiotopoulos, Aristotelis Logic Dynamical Systems Countable $\mathcal{L}$-structures $\mathcal{N}$ whose isomorphism class supports a permutation invariant probability measure in the logic action have been characterized by Ackerman-Freer-Patel to be precisely those $\mathcal{N}$ which have no algebraicity. Here we characterize those countable $\mathcal{L}$-structure $\mathcal{N}$ whose isomorphism class supports a quasi-invariant probability measure. These turn out to be precisely those $\mathcal{N}$ which are not "highly algebraic" -- we say that $\mathcal{N}$ is highly algebraic if outside of every finite $F$ there is some $b$ and a tuple $\bar{a}$ disjoint from $b$ so that $b$ has a finite orbit under the pointwise stabilizer of $\bar{a}$ in $\mathrm{Aut}(\mathcal{N})$. As a bi-product of our proof we show that whenever the isomorphism class of $\mathcal{N}$ admits a quasi-invariant measure, then it admits one with continuous Radon--Nikodym cocycles. |
| title | Quasi-invariant measures concentrating on countable structures |
| topic | Logic Dynamical Systems |
| url | https://arxiv.org/abs/2408.07454 |