Title :

Around Furstenberg's times $p$, times $q$ conjecture: times $p$-invariant measures with some large Fourier coefficients

Author(s) :

Badea, Catalin [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Grivaux, Sophie [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]

Journal title :

Discrete Analysis

Publisher :

Alliance of Diamond Open Access Journals

Publication date :

2024-11-06

ISSN :

2397-3129

English keyword(s) :

$\times_p$-invariant measures
Furstenberg Conjecture
Fourier coefficients of continuous measures
Baire Category methods

HAL domain(s) :

Mathématiques [math]

English abstract : [en]

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 ...
Show more >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 $\mu$ on \(\mathbb{T}\) with no atom has some large Fourier coefficients along the sequence $(q^n)_{n\ge 0}$. In particular, \((T_{q^{n}}\mu )_{n\ge 0}\) does not converges weak-star to the normalised Lebesgue measure on $\mathbb{T}$. This disproves a conjecture of Furstenbergand 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.Show less >

Language :

Anglais

Popular science :

Non

ANR Project :

Centre Européen pour les Mathématiques, la Physique et leurs Interactions

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL