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 on induces an action on the projective space . For later use, recall that via

Given a probability measure on , we can build a Markov chain / random walk whose steps consist into taking points into where is chosen accordingly with the law of .

The absence of hypothesis on might lead to uninteresting random walks: in fact, if a point is stabilized by two elements , then the random walk starting at associated to is not very interesting.

For this reason, we shall assume that

Hypothesis (i): the support of generates a *Zariski-dense* semigroup .

Remark 1By Tits alternative, in our current setting of , 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 have a well-defined asymptotic behaviour whenever (i) is fulfilled:

Theorem 1 (Furstenberg)Under (i), there exists (an unique) probability measure on such that, for all ,

as . Here, the convolution of with a probability measure on is a probability measure on defined as

so that is the distribution of points obtained from after steps of the Markov chain associated to .

In the literature, is called *Furstenberg measure*, and it is an important example of –*stationary measure*, i.e., a probability measure on which is “invariant on average”:

**1.2. Lyapunov exponents**

The stationary measure can be used to describe the growth of the norms of random products associated to -almost every whenever satisfies (i) *and* its first moment is finite:

Theorem 2 (Furstenberg, Guivarch–Raugi)If has finite first moment, i.e.,

and satisfies (i), then

for -almost every .

The quantity is called *Lyapunov exponent*.

**1.3. Regularity of stationary measures**

The Furstenberg measure dictates the distribution of the Markov chains associated to 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 satisfies (i) and its exponential moment is finite:

Hypothesis (ii): there exists with .

Theorem 3 (Guivarch)Under (i) and (ii), there are and such that

for all and (where is the interval of radius centered at ). In particular, has no atoms.

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

(where ). More concretely, he proved that:

Theorem 4 (Li)Under (i) and (ii), we have . In other words, 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 1Let with

Note that the semigroup generated by is not Zariski dense in (as and are upper-triangular).We affirm that there is no decay of Fourier coefficients in this situation. Indeed, recall that if we identify with via , then acts on via Möbius transformations, i.e., an element acts on as

In particular, , , 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 is a real number such that is a Pisot number, then with

admits a stationary measure (called the Bernoulli convolution of parameter describing the distribution of the points where with probability ) whose Fourier coefficients do not decay.

The proof of Theorem 4 is based on a renewal theorem. More concretely, given a function , let

By thinking of as a smooth version of the characteristic function of an interval , we see that is “counting random products with norm in the interval ”. In this context, Guivarch and Le Page established the following renewal theorem:

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

as .

Remark 2Another important fact in the proof of Theorem 4 is the non-arithmeticity of the Jordan projections of the elements of , i.e., the fact that these Jordan projections generate a dense subgroup of (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 is the Patterson–Sullivan measure associated to a Schottky subgroup of , then there exists (depending only on the dimension of , i.e., the Hausdorff dimension of the limit set of the Schottky subgroup) such that

for all .

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 of a Schottky subgroup coincides with the stationary measure of some probability measure on satisfying (i) and (ii).

Remark 3We saw the proof of a version of this result for cocompact lattices of 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 on satisfying (i) and (ii). This statement was recently proved by Jialun Li in this article here.

Theorem 8 (Li)If is a probability measure on satisfying (i) and (ii), then there exists such that the Furstenberg measure associated to verifies

for all .

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

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 be a probability measure on verifying (i) and (ii). Given , consider the transfer operator

acting on (with small enough). Then,we have the following spectral gap property: there exists such that the spectral radius of satisfies

for all .

Theorem 10 (Li)Under (i) and (ii), there exists such that the renewal operator satisfies

for all .

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 8, 9 and 10 are “morally equivalent” to each other. In fact,

- Theorem 8 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 Theorem 10: the spectral gap for transfer operators allows to deduce the renewal theorem because some elementary calculations reveal that is related to ;
- Theorem 10 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 , the th Fourier coefficient of the Furstenberg measure is
by Cauchy–Schwarz inequality, the control of is reduced to the study of

since , we see that the size of the integral above depends on the “number of random products with norm in a given interval”, and the answer to this kind of “counting problem” is encoded in the asymptotic property of the renewal operator provided by Theorem 10.

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

for , but the error term is never exponential because grows polynomially with . (Of course, this phenomenon is avoided in the context of thanks to the fact that the Zariski-density assumption (i) on ensures an exponential growth of with .)

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