Density of orbits of horocycle flows at sub-quadratic polynomial times
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| Format: | Preprint |
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2025
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| _version_ | 1866911231148818432 |
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| author | Kanigowski, Adam Radziwiłł, Maksym |
| author_facet | Kanigowski, Adam Radziwiłł, Maksym |
| contents | Let $Γ\subset PSL(2,\mathbb{R})$ be such that the space $X=Γ\backslash PSL(2,\mathbb{R})$ is not compact. Let $(h_t)$ be the horocycle flow acting on $X$. We show that for every $x\in X$ that is not periodic for $(h_t)$ and for every $δ\in (0,1)$ the orbit $\{h_{n^{2-δ}}x\}_{n\in \mathbb{N}}$ is dense in $X$. Assuming additionally the Hardy-Littlewood conjecture we show that for every non-periodic $x\in X$, $\{h_{p}x\}_{p- \text{prime}}$ is dense in $X$. Finally we show that for $Γ=PSL(2,\mathbb{Z})$, $\{h_{n^2}y_q\}_{n<q}$ equidistribute, as $q\to \infty$ along primes congruent to $1 \pmod{4}$, towards Haar measure, where $\{y_q\}$ is a sequence of periodic points of period $q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_21964 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Density of orbits of horocycle flows at sub-quadratic polynomial times Kanigowski, Adam Radziwiłł, Maksym Dynamical Systems Number Theory 37A15, Let $Γ\subset PSL(2,\mathbb{R})$ be such that the space $X=Γ\backslash PSL(2,\mathbb{R})$ is not compact. Let $(h_t)$ be the horocycle flow acting on $X$. We show that for every $x\in X$ that is not periodic for $(h_t)$ and for every $δ\in (0,1)$ the orbit $\{h_{n^{2-δ}}x\}_{n\in \mathbb{N}}$ is dense in $X$. Assuming additionally the Hardy-Littlewood conjecture we show that for every non-periodic $x\in X$, $\{h_{p}x\}_{p- \text{prime}}$ is dense in $X$. Finally we show that for $Γ=PSL(2,\mathbb{Z})$, $\{h_{n^2}y_q\}_{n<q}$ equidistribute, as $q\to \infty$ along primes congruent to $1 \pmod{4}$, towards Haar measure, where $\{y_q\}$ is a sequence of periodic points of period $q$. |
| title | Density of orbits of horocycle flows at sub-quadratic polynomial times |
| topic | Dynamical Systems Number Theory 37A15, |
| url | https://arxiv.org/abs/2510.21964 |