Posted by: matheuscmss | February 3, 2020

## Decrease of Fourier coefficients of stationary measures on the circle (after Jialun Li)

Last January 8, 2020, Jialun Li gave the talk “Decrease of Fourier coefficients of stationary measures on the circle” in the “flat seminar” that I co-organize with Anton Zorich once per month.

In this post, I’ll transcript my notes of this nice talk (while taking full responsibility for any errors/mistakes in what follows).

1. Introduction

1.1. Stationary measures

Consider the linear action of ${G=SL_2(\mathbb{R})}$ on ${\mathbb{R}^2}$ induces an action on the projective space ${X=\mathbb{P}(\mathbb{R}^2)}$. For later use, recall that ${X\simeq\mathbb{T}^1=\mathbb{R}/\pi\mathbb{Z}}$ via

$\displaystyle \mathbb{T}^1\ni\theta\mapsto [\cos\theta:\sin\theta]=\mathbb{R}\cdot(\cos\theta,\sin\theta)\in X.$

Given a probability measure ${\mu}$ on ${G}$, we can build a Markov chain / random walk whose steps consist into taking points ${y\in X}$ into ${gy}$ where ${g\in G}$ is chosen accordingly with the law of ${\mu}$.

The absence of hypothesis on ${\mu}$ might lead to uninteresting random walks: in fact, if a point ${x\in X}$ is stabilized by two elements ${g,h\in Stab_x(G)}$, then the random walk starting at ${x}$ associated to ${\mu=\frac{1}{2}(\delta_g+\delta_h)}$ is not very interesting.

For this reason, we shall assume that

Hypothesis (i): the support of ${\mu}$ generates a Zariski-dense semigroup ${\langle \textrm{supp}(\mu)\rangle}$.

Remark 1 By Tits alternative, in our current setting of ${G=SL_2(\mathbb{R})}$, the hypothesis (i) can be reformulated by replacing “Zariski-dense” with “not solvable”.

As it was famously established by Furstenberg, the random walks associated to ${\mu}$ have a well-defined asymptotic behaviour whenever (i) is fulfilled:

Theorem 1 (Furstenberg) Under (i), there exists (an unique) probability measure ${\nu}$ on ${X}$ such that, for all ${x\in X}$,

$\displaystyle \mu^n\ast\delta_x\rightharpoonup\nu$

as ${n\rightarrow\infty}$. Here, the convolution of ${\mu}$ with a probability measure ${\eta}$ on ${X}$ is a probability measure ${\mu\ast\eta}$ on ${X}$ defined as

$\displaystyle \mu\ast\eta = \int_{g\in G} g_*\eta \, \, d\mu(g),$

so that ${\mu^n\ast\delta_x=\underbrace{\mu\ast\dots\ast\mu}_{n \textrm{ times}}\ast\nu}$ is the distribution of points obtained from ${x}$ after ${n}$ steps of the Markov chain associated to ${\mu}$.

In the literature, ${\nu}$ is called Furstenberg measure, and it is an important example of ${\mu}$stationary measure, i.e., a probability measure on ${X}$ which is “invariant on average”:

$\displaystyle \nu=\int_{g\in G}g_*\nu \, \, d\mu(g) := \mu\ast\nu.$

1.2. Lyapunov exponents

The stationary measure ${\nu}$ can be used to describe the growth of the norms of random products ${g_n\dots g_1}$ associated to ${\mu^{\otimes\mathbb{N}}}$-almost every ${(g_1,\dots, g_n,\dots)\in G^{\mathbb{N}}}$ whenever ${\mu}$ satisfies (i) and its first moment is finite:

Theorem 2 (Furstenberg, Guivarch–Raugi) If ${\mu}$ has finite first moment, i.e.,

$\displaystyle \int_G\log\|g\|\ d\mu(g)<\infty$

and ${\mu}$ satisfies (i), then

$\displaystyle \lim\limits_{n\rightarrow\infty}\frac{1}{n}\log\|g_n\dots g_1\|= \sigma_{\mu}:=\int_{x=\mathbb{R}v\in X}\int_{g\in G}\log\frac{\|gv\|}{\|v\|} d\mu(g)\,d\nu(x)>0$

for ${\mu^{\otimes\mathbb{N}}}$-almost every ${(g_1,\dots, g_n, \dots)\in G^{\mathbb{N}}}$.

The quantity ${\sigma_{\mu}}$ is called Lyapunov exponent.

1.3. Regularity of stationary measures

The Furstenberg measure dictates the distribution of the Markov chains associated to ${\mu}$ and, for this reason, it is natural to inquiry about the regularity properties of stationary measures.

In this direction, Guivarch showed that the Furstenberg measures have a certain regularity when ${\mu}$ satisfies (i) and its exponential moment is finite:

Hypothesis (ii): there exists ${\theta>0}$ with ${\int_{g\in G} \|g\|^{\theta}\,\,d\mu(g)<\infty}$.

Theorem 3 (Guivarch) Under (i) and (ii), there are ${\alpha>0}$ and ${C>0}$ such that

$\displaystyle \nu(B(x,r))\leq C r^{\alpha}$

for all ${x\in X}$ and ${r>0}$ (where ${B(x,r)}$ is the interval of radius ${r}$ centered at ${x\in X\simeq\mathbb{T}^1}$). In particular, ${\nu}$ has no atoms.

More recently, Jialun Li established in this article here another regularity result by showing the decay of the Fourier coefficients

$\displaystyle \widehat{\nu}(k):=\int_X e^{2ikx}d\nu(x), \quad k\in\mathbb{Z},$

(where ${X=\mathbb{P}(\mathbb{R}^2)\simeq\mathbb{T}^1=\mathbb{R}/\pi\mathbb{Z}}$). More concretely, he proved that:

Theorem 4 (Li) Under (i) and (ii), we have ${\lim\limits_{|k|\rightarrow\infty}\hat{\nu}(k)=0}$. In other words, ${\nu}$ is a Rajchman measure.

In a certain sense, the role of assumption (i) in the previous theorem is to avoid the following kind of example:

Example 1 Let ${\mu=\frac{1}{2}(\delta_g+\delta_h)}$ with

$\displaystyle g=\left(\begin{array}{cc} \frac{1}{\sqrt{3}} & 0 \\ 0 & \sqrt{3}\end{array}\right) \quad \textrm{and} \quad h=\left(\begin{array}{cc} \frac{1}{\sqrt{3}} & \frac{2}{\sqrt{3}} \\ 0 & \sqrt{3}\end{array}\right).$

Note that the semigroup generated by ${\textrm{supp}(\mu)=\{a,b\}}$ is not Zariski dense in ${G}$ (as ${a}$ and ${b}$ are upper-triangular).We affirm that there is no decay of Fourier coefficients in this situation. Indeed, recall that if we identify ${X}$ with ${\mathbb{R}\cup\{\infty\}}$ via ${X\ni x=\mathbb{R}\cdot (v_1,v_2)\mapsto v_1/v_2\in\mathbb{R}\cup\{\infty\}}$, then ${G}$ acts on ${\mathbb{R}\cup\{\infty\}}$ via Möbius transformations, i.e., an element ${\left(\begin{array}{cc} a & b \\ c & d\end{array}\right)\in G}$ acts on ${x\in\mathbb{R}\cup\{\infty\}}$ as

$\displaystyle \left(\begin{array}{cc} a & b \\ c & d\end{array}\right)x=\frac{ax+b}{cx+d}.$

In particular, ${gx=x/3}$, ${hx=(x+2)/3}$, and the Fourier coefficients of the stationary measure given by the standard Hausdorff measure on middle-third Cantor set do not decay to zero.In a similar vein, if ${0<\lambda<1}$ is a real number such that ${1/\lambda}$ is a Pisot number, then ${\mu=\frac{1}{2}(\delta_{g_{\lambda}}+\delta_{h_{\lambda}})}$ with

$\displaystyle g_{\lambda}=\left(\begin{array}{cc} \sqrt{\lambda} & -\frac{1}{\sqrt{\lambda}} \\ 0 & \frac{1}{\sqrt{\lambda}}\end{array}\right) \quad \textrm{and} \quad h=\left(\begin{array}{cc} \sqrt{\lambda} & \frac{1}{\sqrt{\lambda}} \\ 0 & \frac{1}{\sqrt{\lambda}}\end{array}\right)$

admits a stationary measure ${\nu_{\lambda}}$ (called the Bernoulli convolution of parameter ${\lambda}$ describing the distribution of the points ${\sum\limits_{j\geq0}\varepsilon_j \lambda^j}$ where ${\varepsilon_j=\pm1}$ with probability ${1/2}$) whose Fourier coefficients do not decay.

The proof of Theorem 4 is based on a renewal theorem. More concretely, given a function ${f\in C^{\infty}_c(\mathbb{R})}$, let

$\displaystyle Rf(t):=\sum\limits_{n=1}^{\infty} \int f(\log\|g\|-t) d\mu^n(g).$

By thinking of ${f}$ as a smooth version of the characteristic function of an interval ${I\subset \mathbb{R}}$, we see that ${Rf(t)}$ is “counting random products ${g_n\dots g_1}$ with norm in the interval ${\exp(I+t)}$”. In this context, Guivarch and Le Page established the following renewal theorem:

Theorem 5 (Guivarch–Le Page) Under (i) and (ii), one has

$\displaystyle Rf(t)\rightarrow \frac{1}{\sigma_{\mu}}\int_{\mathbb{R}} f(u) \, du$

as ${t\rightarrow\infty}$.

Remark 2 Another important fact in the proof of Theorem 4 is the non-arithmeticity of the Jordan projections ${\lambda(g)=\log (\textrm{top eigenvalue of }g)}$ of the elements of ${\langle \textrm{supp}(\mu)\rangle}$, i.e., the fact that these Jordan projections generate a dense subgroup of ${\mathbb{R}}$ (whenever (i) is satisfied).

Since we will come back later to the discussion of deriving the decay of Fourier coefficients (e.g., Theorem 4) from a renewal theorem, let us now move forward in order to introduce the main result of this post, namely, a quantitative version of Theorem 4.

2. Quantitative decay of Fourier coefficients

The central result of this post is inspired by the following theorem of Bourgain and Dyatlov.

Theorem 6 (Bourgain–Dyatlov) If ${\nu_{PS}}$ is the Patterson–Sullivan measure associated to a Schottky subgroup of ${SL_2(\mathbb{R})}$, then there exists ${\varepsilon>0}$ (depending only on the dimension of ${\nu_{PS}}$, i.e., the Hausdorff dimension of the limit set of the Schottky subgroup) such that

$\displaystyle |\widehat{\nu_{PS}}(k)|=O(|k|^{-\varepsilon})$

for all ${k\in\mathbb{Z}}$.

The method of proof of this result is based on the so-called discretized sum-product estimates from additive combinatorics.

Interestingly enough, this result can be interpreted as a decay of Fourier coefficients of certain stationary measures thanks to the following theorem:

Theorem 7 (Furstenberg, Sullivan, …) The Patterson–Sullivan measure ${\nu_{PS}}$ of a Schottky subgroup coincides with the stationary measure ${\nu}$ of some probability measure ${\mu}$ on ${G}$ satisfying (i) and (ii).

Remark 3 We saw the proof of a version of this result for cocompact lattices of ${SL_2(\mathbb{R})}$ in Proposition 14 of this blog post here.

The previous theorems suggest that a decay of Fourier coefficients of the Furstenberg measure associated to a probability measure ${\mu}$ on ${G}$ satisfying (i) and (ii). This statement was recently proved by Jialun Li in this article here.

Theorem 8 (Li) If ${\mu}$ is a probability measure on ${G=SL_2(\mathbb{R})}$ satisfying (i) and (ii), then there exists ${\varepsilon>0}$ such that the Furstenberg measure ${\nu}$ associated to ${\mu}$ verifies

$\displaystyle |\widehat{\nu}(k)|=O(|k|^{-\varepsilon})$

for all ${k\in\mathbb{Z}}$.

Remark 4 Actually, Li’s theorem is stated in his article for any real split semisimple Lie group ${G}$.

The proof of this result is also based on a discretized sum-product estimate. Moreover, this statement is closely related to spectral gap of transfer operators and a renewal theorem:

Theorem 9 (Li) Let ${\mu}$ be a probability measure on ${G}$ verifying (i) and (ii). Given ${b\in \mathbb{R}}$, consider the transfer operator

$\displaystyle P_{ib} f(x) := \int_G \exp(i b \log\frac{\|gv\|}{\|v\|}) f(gx) \, d\mu(g), \quad x=\mathbb{R}\cdot v\in X,$

acting on ${f\in C^{\gamma}(X)}$ (with ${\gamma>0}$ small enough). Then,we have the following spectral gap property: there exists ${\rho<1}$ such that the spectral radius of ${P_{ib}}$ satisfies

$\displaystyle \rho(P_{ib})<\rho$

for all ${|b|>1}$.

Theorem 10 (Li) Under (i) and (ii), there exists ${\varepsilon>0}$ such that the renewal operator ${Rf(t)=\sum\limits_{n=1}^{\infty} \int f(\log\|g\|-t) d\mu^n(g)}$ satisfies

$\displaystyle Rf(t) = \frac{1}{\sigma_{\mu}}\int_{\mathbb{R}} f(u) \, du + O(e^{-\varepsilon t} |f|_{C^2})$

for all ${f\in C^{\infty}_c(\mathbb{R})}$.

In his article, Li establishes first Theorem 8 from a discretized sum-product estimate, and subsequently Theorems 9 and 10 are deduced from Theorem 8.

Nevertheless, Li pointed out in his talk that Theorems 89 and 10 are “morally equivalent” to each other. In fact,

• Theorem 8 ${\implies}$ Theorem 9: the Fourier decay can be used to prove spectral gap for transfer operators via the so-called Dolgopyat method (which was discussed in this blog post here);
• Theorem 9 ${\implies}$ Theorem 10: the spectral gap for transfer operators allows to deduce the renewal theorem because some elementary calculations reveal that ${Rf(t)}$ is related to ${(Id-P_{ib})^{-1}}$;
• Theorem 10 ${\implies}$ Theorem 8: let us finally fulfil our promise made in the end of the previous section by briefly explaining the idea of the derivation of the Fourier decay in Theorem 8 from the renewal theorem in Theorem 10; since ${\mu^n\ast\nu=\nu}$, the ${k}$th Fourier coefficient of the Furstenberg measure is

$\displaystyle \widehat{\nu}(k) = \int_X e^{2ikx}\,d\nu(x) = \int_G \int_X e^{2ikx}\,d\mu^n(g)\,d\nu(x);$

by Cauchy–Schwarz inequality, the control of ${|\widehat{\nu}(k)|}$ is reduced to the study of

$\displaystyle \int_G e^{2ik(gx-gy)} d\mu^n(g);$

since ${gx-gy\sim \|g\|^{-1} d(x,y)}$, we see that the size of the integral above depends on the “number of random products ${g=g_n\dots g_1}$ with norm ${\|g\|}$ in a given interval”, and the answer to this kind of “counting problem” is encoded in the asymptotic property of the renewal operator ${Rf(t)}$ provided by Theorem 10.

Remark 5 The analog of Theorem 10 in Abelian settings is false: the random walks driven by a finitely supported law ${\lambda}$ on ${\mathbb{R}}$ which is not arithmetic (i.e., its support generates a dense subgroup) verify a renewal theorem

$\displaystyle Rf(t)=\sum\limits_{n=1}^{\infty}\int f(x-t) d\lambda^n(x)\rightarrow \frac{1}{\mathbb{E}(\lambda)}\int f(u) \, du \quad \textrm{as } t\rightarrow\infty$

for ${f\in C^{\infty}_c(\mathbb{R})}$, but the error term is never exponential because ${\#\textrm{supp}(\lambda^n)}$ grows polynomially with ${n}$. (Of course, this phenomenon is avoided in the context of ${SL_2(\mathbb{R})}$ thanks to the fact that the Zariski-density assumption (i) on ${\mu}$ ensures an exponential growth of ${\#\textrm{supp}(\mu^n)}$ with ${n}$.)

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