The influence of the maximal summand on ergodic sums of non-integrable observables over rotations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916916363264000 |
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| author | Kanigowski, Adam Schindler, Tanja I. |
| author_facet | Kanigowski, Adam Schindler, Tanja I. |
| contents | For $R_α$ being an irrational rotation of angle $α$ on the one torus $\mathbb{T}$ and $ϕ(x)=\frac{1}{x}-\frac{1}{1-x}$, we compare the behavior of the Birkhoff sum $S_N(ϕ)=\sum_{k=0}^{N-1}(ϕ\circ R_α^k)(x)$ with the successive entry $(ϕ\circ R_α^N)(x)$. In particular, we are interested in the almost sure limsup behavior of $\frac{(ϕ\circ R_α^N)(x)}{S_N(ϕ)(x)}$. We show that depending on the Diophantine properties of $α$ we have that the limsup either equals $0$ or $\infty$. Moreover, we show that those $α$ for which the limsup equals $0$ form an atypical set in the sense that its Hausdorff dimension equals $\frac{1}{2}$. These results have consequences in studying a reparametrization $(T_t)$ of the linear flow $(L_t)$ with direction $(1,α)$ on the two torus $\mathbb{T}^2$ with function $φ$, where $φ$ is a smooth non-negative function that has exactly two (non-degenerate) zeros at $\bf p$ and $\bf q$. We prove that for a full measure set $(α, {\bf p}, {\bf q})\in \mathbb{T}\times \mathbb{T}^2\times \mathbb{T}^2$ the special flow $(T_t)$ exhibits extreme historic behavior proving a conjecture given by Andersson and Guihéneuf. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_18216 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The influence of the maximal summand on ergodic sums of non-integrable observables over rotations Kanigowski, Adam Schindler, Tanja I. Dynamical Systems Number Theory 60F20, 37A44, 37C10, 37C40, 11K50, 60G70 For $R_α$ being an irrational rotation of angle $α$ on the one torus $\mathbb{T}$ and $ϕ(x)=\frac{1}{x}-\frac{1}{1-x}$, we compare the behavior of the Birkhoff sum $S_N(ϕ)=\sum_{k=0}^{N-1}(ϕ\circ R_α^k)(x)$ with the successive entry $(ϕ\circ R_α^N)(x)$. In particular, we are interested in the almost sure limsup behavior of $\frac{(ϕ\circ R_α^N)(x)}{S_N(ϕ)(x)}$. We show that depending on the Diophantine properties of $α$ we have that the limsup either equals $0$ or $\infty$. Moreover, we show that those $α$ for which the limsup equals $0$ form an atypical set in the sense that its Hausdorff dimension equals $\frac{1}{2}$. These results have consequences in studying a reparametrization $(T_t)$ of the linear flow $(L_t)$ with direction $(1,α)$ on the two torus $\mathbb{T}^2$ with function $φ$, where $φ$ is a smooth non-negative function that has exactly two (non-degenerate) zeros at $\bf p$ and $\bf q$. We prove that for a full measure set $(α, {\bf p}, {\bf q})\in \mathbb{T}\times \mathbb{T}^2\times \mathbb{T}^2$ the special flow $(T_t)$ exhibits extreme historic behavior proving a conjecture given by Andersson and Guihéneuf. |
| title | The influence of the maximal summand on ergodic sums of non-integrable observables over rotations |
| topic | Dynamical Systems Number Theory 60F20, 37A44, 37C10, 37C40, 11K50, 60G70 |
| url | https://arxiv.org/abs/2508.18216 |