On the spectrum of non-ergodic measures
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866909985439481856 |
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| author | Francis, Michael Ramsey, Christopher Strungaru, Nicolae |
| author_facet | Francis, Michael Ramsey, Christopher Strungaru, Nicolae |
| contents | Consider a topological dynamical system where the group is abelian and the topologies are locally compact and second-countable. Given an invariant measure for this system, we show that if its dynamical spectrum is contained in some Borel subset of the dual group then the same holds almost surely for all ergodic measures arising via the Choquet theorem. In particular, if the invariant measure has pure point dynamical spectrum, so do almost all the ergodic measures. As an application, we show that given any mean almost periodic measure, in its hull there exists a Besicovitch almost periodic measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_05327 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the spectrum of non-ergodic measures Francis, Michael Ramsey, Christopher Strungaru, Nicolae Dynamical Systems Functional Analysis Consider a topological dynamical system where the group is abelian and the topologies are locally compact and second-countable. Given an invariant measure for this system, we show that if its dynamical spectrum is contained in some Borel subset of the dual group then the same holds almost surely for all ergodic measures arising via the Choquet theorem. In particular, if the invariant measure has pure point dynamical spectrum, so do almost all the ergodic measures. As an application, we show that given any mean almost periodic measure, in its hull there exists a Besicovitch almost periodic measure. |
| title | On the spectrum of non-ergodic measures |
| topic | Dynamical Systems Functional Analysis |
| url | https://arxiv.org/abs/2601.05327 |