Decrease of Fourier coefficients of stationary measures

Abstract

Let µ be a Borel probability measure on SL 2 (R) with a finite exponential moment, and assume that the subgroup Γ µ generated by the support of µ is Zariski dense. Let ν be the unique µ−stationary measure on P 1 . We prove that the Fourier coefficientsν(k) of ν converge to 0 as |k| tends to infinity. Our proof relies on a generalized renewal theorem for the Cartan projection.Corollary 1.8. Let Γ be a convex cocompact Fuchsian group. Then the Patterson-Sullivan measure associated to Γ is a Rajchman measure. Remark 1.9. Corollary 1.8 also holds if we replace the Patterson-Sullivan measure by any Gibbs measure. In [Li], we have a similar realization for any Gibbs measure associated to a convex cocompact Fuchsian group, as it is done by Lalley for any Gibbs measure on the limit set of a Schottky semigroup in [Lal86].

Decay of Fourier coefficients for Furstenberg measures

“…Under the stronger exponential moment condition, i.e., G R g α dµ(g) < ∞ for some α > 0, the above result is due to Li [Li18]. The sequel [Li20] provides a polynomial decay rate, that is, | ν(k)| 1/|k| β for some β > 0.…”

“…When µ has a finite exponential moment an analogous decomposition holds in some Hölder space. In this case, the situation is considerably better and the family U ξ extends analytically through ξ = 0, see for instance [Boy16,Li18,LS20]. The weak control from Proposition 3.10 makes our analysis more involved.…”

“…The proof of Theorem A follows the strategy of [Li18] and draw from techniques of our recent works [DKW20,DKW21a,DKW21b]. The key idea is to use stopping times and Cauchy-Schwarz inequality in order to bound the integral defining ν(k) by a quantitity that can be translated to fit the framework of renewal theory for Markov chains, as initiated by Kesten [Kes74] and further developed in [Boy16,GLP16,Li18,Li20]. We refer to Section 5 for more details on the structure of the proof.…”

“…The main difficulties are the fact that natural pertubations of the Markov operator display very low regularity (see Section 3) and that the large deviation estimates, as obtained in [BQ16a], are weaker in the finite second moment case. Therefore, in order to be able to apply Li's method, we need to further develop the spectral analysis of the perturbed Markov operators initiated in [DKW21b] and generalize most of the estimates of [Li18] to cover the case of finite second moment. Due to the low regularity of the perturbed Markov operators, new arguments are needed.…”

Logarithmic Fourier decay for self conformal measures

“…The second method is based on renewal theory and was originally developed by Li [22] to study the Rajchman property for Furstenberg measures. Li-Sahlsten [25,Theorem 1.2] then adapted this method to prove that if the contraction ratios of the IFS are {r 1 , ..., r n } ⊂ R + and there exist i, j such that log r i log r j / ∈ Q, then all self-similar measures are Rajchman.…”