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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2303.01089 |
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| _version_ | 1866915006477500416 |
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| author | Badea, Catalin Grivaux, Sophie |
| author_facet | Badea, Catalin Grivaux, Sophie |
| contents | For each integer $n\ge 1$, denote by $T_{n}$ the map $x\mapsto nx\mod 1$ from the circle group $\mathbb{T} = \mathbb{R}/\mathbb{Z}$ into itself. Let $p,q\ge 2$ be two multiplicatively independent integers. Using Baire Category arguments, we show that generically a $T_{p}$-invariant probability measure $μ$ on $\mathbb{T}$ with no atom has some large Fourier coefficients along the sequence $(q^n)_{n\ge 0}$. In particular, $(T_{q^{n}}μ)_{n\ge 0}$ does not converges weak-star to the normalised Lebesgue measure on $\mathbb{T}$. This disproves a conjecture of Furstenberg and complements previous results of Johnson and Rudolph. In the spirit of previous work by Meiri and Lindenstrauss-Meiri-Peres, we study generalisations of our main result to certain classes of sequences $(c_n)_{n\ge 0}$ other than the sequences $(q^{n})_{n\ge 0}$, and also investigate the multidimensional setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_01089 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Around Furstenberg's times $p$, times $q$ conjecture: times $p$-invariant measures with some large Fourier coefficients Badea, Catalin Grivaux, Sophie Dynamical Systems Functional Analysis Number Theory 43A25 (Primary) 37A05, 54E52, 37A25 (Secondary) For each integer $n\ge 1$, denote by $T_{n}$ the map $x\mapsto nx\mod 1$ from the circle group $\mathbb{T} = \mathbb{R}/\mathbb{Z}$ into itself. Let $p,q\ge 2$ be two multiplicatively independent integers. Using Baire Category arguments, we show that generically a $T_{p}$-invariant probability measure $μ$ on $\mathbb{T}$ with no atom has some large Fourier coefficients along the sequence $(q^n)_{n\ge 0}$. In particular, $(T_{q^{n}}μ)_{n\ge 0}$ does not converges weak-star to the normalised Lebesgue measure on $\mathbb{T}$. This disproves a conjecture of Furstenberg and complements previous results of Johnson and Rudolph. In the spirit of previous work by Meiri and Lindenstrauss-Meiri-Peres, we study generalisations of our main result to certain classes of sequences $(c_n)_{n\ge 0}$ other than the sequences $(q^{n})_{n\ge 0}$, and also investigate the multidimensional setting. |
| title | Around Furstenberg's times $p$, times $q$ conjecture: times $p$-invariant measures with some large Fourier coefficients |
| topic | Dynamical Systems Functional Analysis Number Theory 43A25 (Primary) 37A05, 54E52, 37A25 (Secondary) |
| url | https://arxiv.org/abs/2303.01089 |