Any minimal system on the circle is either uniquely ergodic or non-statistical
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910272480870400 |
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| author | Sarizadeh, Aliasghar |
| author_facet | Sarizadeh, Aliasghar |
| contents | This note establishes a connection between topological dynamics and statistical properties in ergodic theory. We begin by demonstrating that systems exhibiting the minimal oscillation property with respect to the Lebesgue measure m (that is, where the sequence of averages of the pullback of m accumulates on at least two distinct measures) are non-statistical. Consequently, this yields a sharp dichotomy: any minimal system on the circle is either uniquely ergodic or non-statistical with respect to the Lebesgue measure. As a result, we conclude that the set of minimal non-statistical systems with respect to Lebesgue measure is non-empty. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_31451 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Any minimal system on the circle is either uniquely ergodic or non-statistical Sarizadeh, Aliasghar Dynamical Systems This note establishes a connection between topological dynamics and statistical properties in ergodic theory. We begin by demonstrating that systems exhibiting the minimal oscillation property with respect to the Lebesgue measure m (that is, where the sequence of averages of the pullback of m accumulates on at least two distinct measures) are non-statistical. Consequently, this yields a sharp dichotomy: any minimal system on the circle is either uniquely ergodic or non-statistical with respect to the Lebesgue measure. As a result, we conclude that the set of minimal non-statistical systems with respect to Lebesgue measure is non-empty. |
| title | Any minimal system on the circle is either uniquely ergodic or non-statistical |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2605.31451 |