Fourier transform and expanding maps on Cantor sets (Q6582301)

One may consult the survey paper by \textit{S.~Dyatlov} [J. Math. Phys. 60, No.~8, 081505, 31~p. (2019; Zbl 1432.81034)] to learn about the background and concepts related to various versions of the Fractal Uncertainty Principle. In this paper, the authors establish polynomial Fourier decay for all Gibbs measures $\mu$ for completely non-linear expanding interval maps $T$ and examine several facets of the Fractal Uncertainty Principle. The following primary result is proved:\N\NTheorem 1.1. Let \(\mathcal{I}_1, \mathcal{I}_2,\dots , \mathcal{I}_N\), \(N\geq 2 \) be closed intervals in \([0,1]\) with disjoint interiors, \( \mathcal{A}=\{1,2,\dots,N\}\) and let \(\{f_a: a \in \mathcal{A}\}\) be a self-conformal iterated function system. Suppose \( {T} : \bigcup_{a=1}^ { N } \mathcal{I}_{a} \rightarrow [0,1]\) is a totally non-linear uniformly expanding Markov map of bounded distortion and \(\mu\) is a non-atomic equilibrium state associated to a potential with exponentially vanishing variations.\N\begin{itemize}\N\item[1.] If \( \bigcup_{a=1}^ { N } \mathcal{I}_{a} = \left[ 0,1\right] \) and \(f_a\) are \(C^2\), then the Fourier coefficients of \(\mu\) tend to zero with a polynomial rate.\N\item[2.] If \(\{ \mathcal{I}_{a} : a \in \mathcal {A} \}\) are disjoint and \(f_a\) are analytic, then the Fourier coefficients of \(\mu\) tend to zero with a polynomial rate.\N\end{itemize}\NTheorem 1.1 is further examined and contrasted with other findings proved in this regard in other contexts [\textit{J.~Bourgain} and \textit{S.~Dyatlov}, Geom. Funct. Anal. 27, No.~4, 744--771 (2017; Zbl 1421.11071); \textit{T.~Jordan} and \textit{T.~Sahlsten}, Math. Ann. 364, No.~3--4, 983--1023 (2016; Zbl 1343.42006)]. There is additional discussion on several other scenarios in which Theorem~1.1 yields certain known findings. Also, it is emphasized that Theorem~1.1 is applicable to a wide range of circumstances, including the ones covered in this article. For new researchers working in this field, this article can be a helpful resource.