Rigorous approximation of diffusion coefficients for expanding maps

We use Ulam's method to provide rigorous approximation of diffusion coefficients for uniformly expanding maps. An algorithm is provided and its implementation is illustrated using Lanford's map.

In [26], following the approach of [14], Pollicott used a Fourier approximation scheme to estimate diffusion coefficients for expanding maps. The approach of [26] requires the map to have a Markov partition and to be piecewise analytic. Although the result of [26] provides an order of convergence, it does not compute the constant hiding in the rate of convergence. In our approach, we do not require the map to admit a Markov partition and we only assume it is piecewise C 2 . More importantly, our approximation is rigorous; i.e., given a map, an observable, and a pre-specified tolerance on error, we approximate the diffusion coefficient rigorously up to the per-specified error (see Theorem 2.3).
In Section 2, we first introduce our system and the assumptions on it. We then state the problem and introduce the method of approximation. The statement of the main result (Theorem 2.3) and the application to expanding maps with a neutral fixed point are also included in Section 2. Section 3 contains the proofs and an algorithm.

The setting
2.1. The system and its transfer operator. Let (I, B, m) be the measure space, where I := [0, 1], B is Borel σ-algebra, and m is the Lebesgue measure on I. Let T : I → I be piecewise C 2 and expanding (see [17,26] for original references 1 and [5] for a profound background on such systems). The transfer operator (Perron-Frobenius) [4] associated with T , P : L 1 → L 1 is defined by duality: for f ∈ L 1 and g ∈ L ∞ Moreover, for f ∈ L 1 we have We denote by BV the space of functions of bounded variation on I equipped with the norm || · || BV = V (·) + || · || 1 . Further, we introduce the mixed operator norm which will play a key role in our approximation:
Remark 2.1. It is important to remark that the constants α and B 0 in (A1) depend only on the map T and have explicit analytic expressions (see [17]).
The above assumptions imply that T admits a unique absolutely continuous invariant measure ν, such that dν dm := h ∈ BV . Moreover, the system (I, B, ν, T ) is mixing and it enjoys exponential decay of correlations for observables in BV (see [4] for a profound background on this topic). 1 In our work, we do not differentiate between maps with finite number of branches [17] or countable (infinite) number of branches [26]. All what we need is a setting where assumptions (A1) and (A2) are satisfied. In fact, using these assumptions, this work can be extended to the multidimensional case [19] by taking care of the dimension [20] and by working with appropriate observables since the space of functions of bounded variations in higher dimension is not contained in L ∞ .
2 It is well known that the systems under consideration satisfy a Lasota-Yorke inequality. What we are assuming in (A1) is that there is no constant in front of α. Such an assumption is satisfied for instance when infx |T ( ′ x)| > 2 or when T is piecewise onto. Also, with the spectral picture provided by (A2), one does not need to state (A1) at all. In stead one can work with a sufficiently high iterate of T where (A1) is automatically satisfied, and then construct the Ulam approximation using the sufficiently high iterate of T . At any rate, it is well know that such a stronger Lasota-Yorke inequality is needed when dealing with Ulam's approximation (see [20]).

The problem.
Let ψ ∈ BV and define Under our assumptions (see [9]) the limit in (2.1) exists, and by using the duality property of P , one can rewrite σ 2 as The number σ 2 is called the variance, or the diffusion coefficient, of . In particular, for the systems under consideration, it is well known (see [9]) that σ 2 > 0 and the Central Limit Theorem holds: The goal of this paper is to provide an algorithm whose output approximates σ 2 with rigorous error bounds. The first step in our approach will be to discretize P as follows: k=1 be a partition of [0, 1] into intervals of size λ(I k ) ≤ ε. Let B η be the σ-algebra generated by η and for f ∈ L 1 define the projection P ε , which is called Ulam's approximation of P , is finite rank operator which can be represented by a (row) stochastic matrix acting on vectors in R d(η) by left multiplication. Its entries are given by The following lemma collects well known results on P ε . See for instance [20] for proofs of (1)-(4) of the lemma, and [20,11] and references therein for statement (5) of the lemma.
2.5. Statement of the main result. Definê 3 says that given a pre-specified tolerance on error τ > 0, one finds l * > 0 and ε * > 0 so that σ 2 ε * ,l * approximates σ up to the pre-specified error τ . In subsection 3.1 we provide an algorithm that can be implemented on a computer to find l * and ε * , and consequently σ 2 ε * ,l * . To illustrate the issue of the rate of convergence and to elaborate on why we define the approximate diffusion by σ 2 ε,l as a truncated sum, let us define Theorem 2.5. ∃ a computable constantK * such that |σ 2 ε − σ 2 | ≤K * ε(ln ε −1 ) 2 . Remark 2.6. Note that σ 2 ε can be written as Since P ε has a matrix representation, and consequently (I − P ε ) −1 is a matrix, one may think that σ 2 ε provides a more sensible formula to approximate σ 2 than σ 2 ε,l . However, from the rigorous computational point of view one has to take into account the errors that arise at the computer level when estimating (I − P ε ) −1 . Indeed (I − P ε ) −1 can be computed rigorously on the computer by estimating it by a finite sum plus an error term coming from estimating the tail of the sum 3 . This is what we do in Theorem 2.3.
Remark 2.7. By using the representation (2.3) of σ 2 ε , it is obvious that the main task in the proof of Theorem 2.5 is to estimate 3 Of course, usual computer software would give an estimated matrix of (I − Pε) −1 , but it does not give the errors it made in its approximation.
where BV 0 = {f ∈ BV s.t. f dm = 0}. Thus, it would be tempting to use estimate (9) in Theorem 1 of [15]. On the one hand, this would lead to a shorter proof than the one we present in section 3; however, estimate (9) of [15] would lead to a convergence rate of order ε θ , where 0 < θ < 1 which is slower than the rate obtained in Theorem 2.5. Obviously, this have led us to opt for using the proofs of section 3.
2.6. Approximating the diffusion coefficient for maps with a neutral fixed point. We now show that Theorem 2.3 can be used to approximate the diffusion coefficient of expanding maps with a neutral fixed point. We restrict the presentation to the model that was popularized by Liverani-Saussol-Vaienti [22]: where the parameter γ ∈ (0, 1). S has a neutral fixed point at x = 0. It is well known that S admits a unique absolutely continuous probability measureν, and the system enjoys polynomial decay of correlation for Hölder observables [29]. For γ ∈ (0, 1 2 ) it is known that the system satisfies the Central Limit Theorem 4 . To study such systems it is often useful to first induce S on a subset of I where the induced map T is uniformly expanding. In particular for the map (2.4), denoting its first branch by S 1 and the second one by S 2 , one can induce S on ∆ := [ 1 2 , 1]. For n ≥ 0 we define W 0 := (x 0 , 1), and W n := (x n , x n−1 ), n ≥ 1. For n ≥ 1, we define Z n := S −1 2 (W n−1 ). Then we define the induced map T : ∆ → ∆ by (2.5) T (x) = S n (x) for x ∈ Z n .
Observe that S(Z n ) = W n−1 and R Zn = n, where R Zn is the first return time of Z n to ∆. For x ∈ ∆, we denote by R(x) the first return time of x to ∆. Let f be Hölder with I f dν = 0. Then diffusion coefficient of the system S can be written using the data of the induced map T (see [13]). In particular, for x ∈ ∆, writing ψ(x) = , the diffusion coefficient is given by where h is the unique invariant density of induced map T , P is the Perron-Frobenius operator associated with T , and m ∆ is normalized Lebesgue measure on ∆. Thus, for ψ ∈ BV one can use 5 , Theorem 2.3 to approximate σ 2 .
4 See [29] for this result and [13] for a more general result. 5 Although T has countable (infinite) number of branches, one can still implement the approximation on a computer. One way to do so is as follows: first one may perform an intermediate step by considering a mapT identical to T on I \ H, such thatT has finite number of branches on I \ H while on H it has, say, one expanding linear branch, with m(H) ≤ δ and δ τ is sufficiently
small. The diffusion coefficients of T andT can be made arbitrarily close using the result of [16], and then one can apply Ulam's method and Theorem 2.3 toT .
We have We know estimate (II): Finally we estimate (III)  We start with (III). Since Iψ hdm = 0, there exists a computable constant C * and a computable number 6 ρ * , where α < ρ * < 1, such that Consequently, Thus, choosing l * such that Fix l * as in (3.5). Now using Lemmas 2.2, 3.1 and 3.2, we can find ε * such that This completes the proof of the theorem.