TITLE: Fourier transform of x2 invariant measure
QUESTION [9 upvotes]: Let $T:\mathbb{R}/\mathbb{Z}\rightarrow \mathbb{R}/\mathbb{Z}$ be the map defined by $T(x)=2x$, and suppose that $\mu$ is a $T$ invariant and ergodic Borel probability measure on the space, which is not a finite sum of point masses. What if anything can be said about the decay of the Fourier transform of $\mu$, along the sequence of odd integers (to take into account Andreas's comment below)?
For example is it true that $\hat{\mu}(2n+1)\ll |2n+1|^{-c}$ for some $c>0$?
If it helps to answer the question (or in case the answer is no in general) I would also be happy to know, what can be said under the additional hypothesis that $T$ has positive measure theoretic entropy with respect to $\mu$?

REPLY [4 votes]: There was an interesting paper posted on the arxiv this morning, Invariant Measures on the Circle and Funcitonal Equations, by Chris Deninger, which seems to essentially answer my question. Corollary 9 of the paper states that a sequence of complex numbers $(c_m)_{m\in\mathbb{Z}}$ is the sequence of Fourier coefficients of a (uniquely determined) times-$n$ invariant special measure (defined below) if and only if:

$c_{-m}=\overline{c}_m$ for all $m\in\mathbb{Z}$,
$c_{mn}=c_m$ for all $m\in\mathbb{Z}$, and
the series $$\sum_{m=0}^\infty |b_m|^2$$ converges, where $(b_m)$ is defined by
$$\sum_{m=0}^\infty b_mz^m=\exp\left(-c_0-2\sum_{m=1}^\infty c_mz^m\right).$$

A finite signed Borel measure $\mu$ is a special measure if $\mu=\alpha \lambda +\nu$ where $\alpha\in\mathbb{R},~\lambda$ is Lebesgue measure, and $\nu$ is a positive singular Borel measure. So in particular any ergodic times-$n$ invariant measure is a special measure and the result above applies.