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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 $\mu$? \mu$, along the sequence of odd integers (to take into account Andreas's comment below)? For example is it true that$\hat{\mu}(n)\ll \hat{\mu}(2n+1)\ll |n|^{-c}$2n+1|^{-c}$ 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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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 discrete measurefinite sum of point masses. What if anything can be said about the decay of the Fourier transform of $\mu$? For example is it true that $\hat{\mu}(n)\ll |n|^{-c}$ 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 we say anything if we assume be said under the additional hypothesis that $T$ has positive measure theoretic entropy with respect to $\mu$?