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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2506.20797 |
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| _version_ | 1866912450367979520 |
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| author | Niemiec, Piotr |
| author_facet | Niemiec, Piotr |
| contents | The main aim of the paper is to introduce a new class of (semigroup-valued) measures that are ultrahomogeneous on the Boolean algebra of all clopen subsets of the Cantor space and to study their automorphism groups. A characterisation, in terms of the so-called trinary spectrum of a measure, of ultrahomogeneous measures such that the action of their automorphism groups is (topologically) transitive or minimal is given. Also sufficient and necessary conditions for the existence of a dense (or co-meager) conjugacy class in these groups are offered. In particular, it is shown that there are uncountably many full non-atomic probability Borel measures m on the Cantor space such that m and all its restrictions to arbitrary non-empty clopen sets have all the following properties: this measure is ultrahomogeneous and not good, the action of its automorphism group G is minimal (on a respective clopen set), and G has a dense conjugacy class. It is also shown that any minimal homeomorphism h on the Cantor space induces a homogeneous h-invariant probability measure that is universal among all h-invariant probability measures (which means that it `generates' all such measures) and this property determines this measure (in a certain sense) up to a Q-linear isomorphism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20797 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Automorphism groups of measures on the Cantor space. Part II: Abstract homogeneous measures Niemiec, Piotr Dynamical Systems Primary 37A15, 28D15, Secondary 28D05, 03E15 The main aim of the paper is to introduce a new class of (semigroup-valued) measures that are ultrahomogeneous on the Boolean algebra of all clopen subsets of the Cantor space and to study their automorphism groups. A characterisation, in terms of the so-called trinary spectrum of a measure, of ultrahomogeneous measures such that the action of their automorphism groups is (topologically) transitive or minimal is given. Also sufficient and necessary conditions for the existence of a dense (or co-meager) conjugacy class in these groups are offered. In particular, it is shown that there are uncountably many full non-atomic probability Borel measures m on the Cantor space such that m and all its restrictions to arbitrary non-empty clopen sets have all the following properties: this measure is ultrahomogeneous and not good, the action of its automorphism group G is minimal (on a respective clopen set), and G has a dense conjugacy class. It is also shown that any minimal homeomorphism h on the Cantor space induces a homogeneous h-invariant probability measure that is universal among all h-invariant probability measures (which means that it `generates' all such measures) and this property determines this measure (in a certain sense) up to a Q-linear isomorphism. |
| title | Automorphism groups of measures on the Cantor space. Part II: Abstract homogeneous measures |
| topic | Dynamical Systems Primary 37A15, 28D15, Secondary 28D05, 03E15 |
| url | https://arxiv.org/abs/2506.20797 |