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 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.
Categories: math.DS math.FA math.NT