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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 $\mu$?

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If I understand correctly, then $\hat \mu(n)= \hat \mu(2n)$ for all $n \in \mathbb N$. There cannot be any decay, unless $\mu = \lambda$. –  Andreas Thom Nov 21 '11 at 21:36
    
@Andreas: Oh, yes of course you are right. I tried to make the statement of the question as general as possible, but I should have caught that. Actually I am only interested in subsequences of odd integers, and I have edited the problem to reflect this. Please let me know if you see any other obvious obstructions. –  Alan Haynes Nov 22 '11 at 12:10
    
This is not exactly what you've asked for, but I think you might want to look in Lemma 3.5 (especially 3.5b) and Lemma 3.6 in this paper by Bourgain and Lindenstrauss - ma.huji.ac.il/~elon/Publications/Effective_Furst.pdf –  Asaf Nov 29 '11 at 17:40
    
@Asaf Thank you. Actually I have read that paper and you are right that there is a relevant idea in there (and in older papers like Host's from the references). The idea is basically that if you assume positive entropy then you can obtain cancellation by averaging over a large collection of Fourier coefficients, if the integers attached to them are well distributed modulo high powers of $n$ ($2$ in my problem). Unfortunately for the problem I have in mind I need more information, which is the motivation for my question. –  Alan Haynes Nov 29 '11 at 22:06
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2 Answers

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:

  1. $c_{-m}=\overline{c}_m$ for all $m\in\mathbb{Z}$,
  2. $c_{mn}=c_m$ for all $m\in\mathbb{Z}$, and
  3. 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.

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As Andreas Thom correctly points out, any invariant Borel probability measure $\hat\mu$ must satisfy $\hat\mu(n)=\hat\mu(2n)$ for all $n \in \mathbb{Z}$, so if the Fourier coefficients tend to zero in the limit then all of them except $\hat\mu(0)$ must be identically zero.

There are at least some additional constraints on the limiting behaviour of the Fourier coefficients of an invariant measure. Since Lebesgue measure is an ergodic measure for $T$, there can be no other invariant measures which are absolutely continuous with respect to Lebesgue. (This can be proved either by showing that the density must be invariant and therefore constant, or by using the Birkhoff ergodic theorem to obtain a contradiction.) While this clearly further constrains the behaviour of the Fourier coefficients, my grasp of Fourier analysis isn't strong enough for me to be able to describe the effect on the coefficients in exact terms.

Any general statement about the behaviour of Fourier coefficients for positive-entropy invariant measures of the doubling map will be constrained by the fact that the set of all such measures is extremely large: every ergodic measure-preserving transformation of a probability space with entropy less than $\log 2$ can be realised as the doubling map equipped with some invariant measure.

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@Ian: Can you explain why polynomial decay of the Fourier transform implies absolute continuity of the measure? I am slightly wary of this, because for example the Kaufman measures have polynomial decay of Fourier coefficients but they are supported on sets of Lebesgue measure $0$... maybe I am missing something. –  Alan Haynes Nov 22 '11 at 12:30
    
Okay. In that case my understanding is incorrect, and what I wrote is wrong; I'll edit that paragraph accordingly. Out of interest, where can I read about Kaufman measures? –  Ian Morris Nov 22 '11 at 12:36
    
The original reference is "Continued fractions and Fourier transforms", R. Kaufman. –  Alan Haynes Nov 22 '11 at 12:47
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