A PROBABILISTIC APPROACH TO INTERMITTENCY

Carlangelo Liverani, Benoît Saussol, Sandro Vaienti

University of Rome Tor Vergata, Centre de Physique Théorique, Marseilles, PhyMat, University of Toulon.

October 3, 1997

Abstract. We present an original approach which allows to investigate the statistical properties of a non-uniform hyperbolic map of the interval. Based on a stochastic approximation of the deterministic map, this method gives essentially the optimal polynomial bound for the decay of correlations, the degree depending on the order of the tangency at the neutral fixed point.

CONTENT 0. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 1 1. The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 2 2. An Invariant Cone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 3 3. A Random Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 5 4. Decay of Correlations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 9 5. General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 13

§0 Introduction. Recently the study of the convergence to the equilibrium in hyperbolic systems has witnessed several new results ranging from new methods to treat systems with singularities [Li1], [Yo] or with partially hyperbolic behavior [BY], to methods for studying Anosov flows [Ch2], [Do], [Li2].

Thanks to such results we can now regard the study of the decay of correlations for uniformly hyperbolic systems as reasonably understood (albeit there is much room for improvements, especially as flows and dependence on smoothness of observable are concerned). On the contrary the available results on the convergence to the equilibrium in non-uniform hyperbolic systems are extremely unsatisfactory.

Typeset by AMS-TEX

We gratefully acknowledge many interesting discussions with V. Baladi, X. Bressaud, V. Maume, B. Schmitt and G. Keller. One of us (B.S.) acknowledges partial support of CEE grant CHRX-CT94-0460. Finally, it is a pleasure to thank the ESI, Vien, and the Institute for Advanced Studies, Jerusalem, where part of this work was done.

The study of such systems stands as a challenge. In particular, it is evident the need to develop new strategies to investigate such problems.

This is the focus of the present paper where we study a one parameter family of intermittent maps. These applications are expanding, except at a neutral fixed point, where hyperbolicity is lost. The local behavior of the map at this point is responsible for various phenomenon. Let us denote by 1 + α the order of the tangency at the critical point. For α = 0 we have a purely expanding map, which has a unique equilibrium state for the potential φ = − log DT, with exponential decay of correlations. For 0 < α < 1, the map possess an absolutely continuous probability measure (SRB measure), which is still an equilibrium state (it is no more unique, since the Dirac mass at the origin is invariant and δ_{0}(φ) = 0). For α ≥ 1, there are no absolutely continuous invariant probability measure, whereas one still has a σ-finite absolutely invariant measures [PS].

We focus here on the second region of the parameter, and propose to find the density of the invariant measure, and the rate of decay of the correlation functions. In this domain, one cannot expect a spectral gap for the Perron-Frobenius operator (see the end of section 4), therefore none of the usual strategies in this setting can be followed. Our approach is based on the following philosophy : sure, the map is not hyperbolic, but it is the case nearly everywhere; thus, if we perform a random perturbation of the map, the neutral fixed point should be lost in a cloud of hyperbolic points and the intermittent effect could be suppressed. This naive argument, rather surprisingly, works.

An interesting property of such a method is the following. For smooth expanding maps, the same idea can be carried out, but yields a sub-exponential rate of decay, while the decay is well known to be exponential. On the contrary, in the present case, the power law found appears to be near optimal, as remarked in section 4.

This is an indication that our crude approach performs better in the non-uniform case than in the uniform one. These considerations are at the base of our belief that this type of strategy could yield relevant results in more general situations.

The plan of the paper is as follows: In section one we present our model and discuss some related literature. Section two is devoted to the study of the invariant measure. The section may have an interest in itself since it gives a very direct approach to obtaining the invariant measure and its properties for such a map (for a comparison with other techniques see [CF,Th]). Section three introduces the key idea of the paper, that is the random perturbation and its instrumental properties. In section four we harvest the facts from the previous sections and obtain the announced result. In addition, we point out that our results suffice to establish the CLT for C^{(1)} observable, provided that α < 1/2 (Remark 4.2-(3)). The last section contains few considerations on how to treat the general problem of expanding maps with neutral fixed points. Since the focus of the present paper is on the method and not on the class of one-dimensional maps to which it can be applied, we content ourselves with few pointed, but sketchy, considerations.

§1 The model. Let us consider for 0 < α < 1 the map T : [0, 1] → [0, 1]

T (x) = \{ \begin{matrix}x(1 + 2^{α}x^{α}) ∀x ∈ [0, 1/2) \\ 2x − 1 ∀x ∈ [1/2, 1] \end{matrix}

The importance of this kind of intermittent maps was addressed by Prellberg and Slawny in [PS], where the relationship with a statistical model introduced by Fisher [FF] and successively studied by Gallavotti [Ga] was emphasized. In the papers [GW], [W], the dynamical behavior of these maps was taken as a model for the intermittency of turbulent flows [PM]. In the paper [PS] several mathematical results were announced, concerning the ergodic and statistical properties of such maps, notably the presence of phase transitions for the topological pressure (see also [Lo]). The paper [W] deals with a piecewise linear version of the map and focus especially on the recurrence properties of the orbits. These latter properties have been put on a solid mathematical basis by Collet, Galves and Schmitt in [CGS] in the piecewise linear case, and by Campanino and Isola for the non-linear (σ-finite) case [CI1,CI2,CI3]. The problem of the decay of correlations was considered in the piecewise linear case and for the finite absolutely continuous (w.r.t. Lebesgue) measure in [LSV] and [Mo]. Both papers obtain an algebraic (n^{−γ}) upper bound for the decay of correlations, the first by using Markov approximations [some results were successively improved by Chernov in [Ch1]], the second by exhibiting the absence of spectral gap for the Perron-Frobenius operator using the induction procedure, already invoked in [PS] (it is interesting to remark that Mori’s work follows the analysis of these maps carried out by Takahashi in a series of papers [Ta1,Ta2]).

The decay of correlations for the non-linear case is a more difficult problem. In our knowledge, the following methods have been proposed.

In the paper [Yu], Yuri applied Markov approximations, generalizing the works of [LSV] and [Ch1]; the paper [Is] aims to extend to the non-linear case the approach of Mori, still inducing, and gives a description of the zeta function, and finally in [FS] the authors propose an interesting technique based on Hilbert metrics, yet the implementation of such an idea is still incomplete.

Let us go back to our particular model. In the next section, we will prove that there exists^{1} a locally Lipschitz function h ∈ C^{(0)}(]0, 1]) ∩ L^{1}([0, 1]) such that P h = h.

§2 An invariant Cone.

If we define the cone C_{0} = {f ∈ C^{(0)}(]0, 1]) | f ≥ 0, f is decreasing} it is immediate to see that C_{0} is left invariant by the P-F operator. To see a bit more let us call X the identity, X(x) = x.

Lemma 2.1. The cone C_{1} = {f ∈ C_{0} | X^{α+1}f is increasing}, is left invariant by the operator P.

Proof. Let f ∈ C_{1}, then

x^{α+1}P f(x) = ∑_{y∈T^{−1}x} (\frac{T y}{y} )^{α+1} \frac{y^{α+1}f(y)}{D_{y}T}.

Setting T^{−1}x = {y_{1}, y_{2}}, y_{1} ≤ y_{2}, and ξ = 2^{α}y_{1}^{α} we can write

x^{α+1}P f(x) = \frac{(1 + ξ)^{α+1}}{1 + (α + 1)ξ} y_{1}^{α+1}f(y_{1}) + \frac{1}{2} (\frac{2y_{2} − 1}{y_{2}})^{α+1} y_{2}^{α+1}f(y_{2}).

1Such a result can also be obtained by inducing [Th]. The method used here is more direct and provides additional informations on the properties of the invariant density, although such extra informations will not be essential in the following. See section five for a more detailed discussion.

4 CARLANGELO LIVERANI, BENOÎT SAUSSOL, SANDRO VAIENTI

Whence the result, since X^{α+1}f, x ↦ y_{1}, x ↦ ξ, x ↦ y_{2} are increasing. □

Let us define

m(f) = ∫_{0}^{1} f(x)dx.

Obviously m(P f) = m(f).

The last interesting property is contained in the following. Lemma 2.2. The cone C_{∗} = {f ∈ C_{1} ∩ L^{1}([0, 1]) | f(x) ≤ ax^{−α}m(f)} is invariant with respect to the operator P, provided a is chosen large enough. Proof. For each f ∈ C_{∗} holds both

f(x) ≤ ax^{−α}m(f),

and

x^{α+1}f(x) ≤ f(1) ≤ ∫_{0}^{1} f = m(f).

Let us suppose for simplicity that m(f) = 1. One has to find a constant a, independent of f, such that P f(x) ≤ ax^{−α} since m(P f) = m(f) = 1.

\begin{matrix}P f(x) = & \frac{f(y_{1})}{D_{y_{1}}T} + \frac{f(y_{2})}{D_{y_{2}}T} \\ ≤ & \frac{ay_{1}^{−α}}{D_{y_{1}}T} + \frac{y_{2}^{−α−1}}{D_{y_{2}}T} \\ ≤ & \{(\frac{x}{y_{1}} )^{α} \frac{1}{D_{y_{1}}T} + \frac{1}{a} \frac{x^{α}}{y_{2}^{α+1}D_{y_{2}}T} \} ax^{−α}. \end{matrix}

The term in curly bracket is bounded by^{2}

\frac{(1 + ξ)^{α}}{1 + (α + 1)ξ} + \frac{2^{α}}{a} ξ ≤ \frac{1 + αξ + \frac{2^{α}}{a} ξ(1 + (α + 1)ξ)}{1 + (α + 1)ξ} ≤ \frac{1 + (α + \frac{2^{α}(α+2)}{a}) ξ}{1 + (α + 1)ξ} ≤ 1.

Whenever a ≥ 2^{α}(α + 2), from which the Lemma follows. □

Putting together all the previous estimates yields Lemma 2.3. There exists a locally Lipschitz function h such that P h = h and h(x) ≤ ax^{−α}. Proof. The operator P leaves invariant the set K = {f ∈ C_{∗} | m(f) = 1}. But X^{α+1}K consists of equibounded equicontinuous functions,^{3} hence it is compact

\begin{aligned}0 ≤ ϕ(x) − ϕ(y) & ≤ (x^{1+α} − y^{1+α})f(x) ≤ a(1 + α)x^{−α} ∫_{y}^{x} ξ^{α}dξ \\ & ≤ a(1 + α)|x − y|. \end{aligned}

2Since \frac{x^{α}}{y_{2}^{α+1}D_{y2} T} ≤ \frac{y_{1}^{α}(1+ξ)^{α}}{2^{−α−1}2} ≤ ξ(1 + ξ)^{α} ≤ 2^{α}ξ.

3Let f ∈ K and define ϕ(x) = x^{1+α}f(x), then, for x ≥ y, holds

in C^{(0)}. Accordingly, for each f ∈ K the sequence X^{α+1} \frac{1}{n} ∑_{i=0}^{n−1} P^{i}f has accumulations points in C^{(0)}. Let h_{∗} ∈ K be such an accumulation point. Clearly, h = X^{−1−α}h_{∗} is a fixed point of P, hence the result. The regularity of h is easily obtained by checking that h ∈ C_{∗}. □

Before introducing the random approximation to our dynamics let us remark a property of the functions in C_{∗} that will be instrumental in the following.

Lemma 2.4. For each function f ∈ C_{∗}

inf_{x∈[0, 1]} f(x) = f(1) ≥ min \{a, [\frac{α(1 + α)}{a^{α}}]^{\frac{1}{1−α}} \} ∫_{0}^{1} f.

Proof. It clearly suffices to consider the case ∫_{0}^{1} f = 1. We have already seen that

\begin{aligned}f(x) & ≤ ax^{−α}, \\ f(x) & ≤ x^{−1−α}f(1). \end{aligned}

We introduce the point x_{∗} = a^{−1}f(1). On its left the first inequality is stricter and the opposite holds on its right. If x_{∗} > 1, then f(1) > a, otherwise

1 = ∫_{0}^{1} f = ∫_{0}^{x_{∗}} f + ∫_{x_{∗}}^{1} f ≤ ∫_{0}^{x_{∗}} aξ^{−α} + ∫_{x_{∗}}^{1} ξ^{−1−α}f(1) ≤ \frac{a^{α}}{α(1 − α)}f(1)^{1−α},

from which the lemma follows. □

§3 A Random Perturbation.

For simplicity let us identify [0, 1] with the circle S^{1}, on S^{1} the map is not smooth but it is continuous (this is not essential but it will make our life a bit easier). Let us define the “ball” B_{ε}(x) = {y ∈ S^{1} | |x − y| ≤ ε} and the averaging operator^{4}

𝔸_{ε}f(x) = \frac{1}{2ε} ∫_{B_{ε}(x)} f(y)dy.

It is now possible to define the perturbed operator

ℙ_{ε} = P^{n_{ε}}𝔸_{ε},

where n_{ε} ∈ ℕ will be specified later.

The following Lemma shows that the perturbed operator is not too different from the original one, provided we consider observables in C_{∗}.

4Let us remark that this particular choice of 𝔸_{ε} has nothing special, any other “reasonable”

choice would do as well.

6 CARLANGELO LIVERANI, BENOÎT SAUSSOL, SANDRO VAIENTI

Lemma 3.1. For each f ∈ C_{∗}

∥P^{n_{ε}}f − ℙ_{ε}f∥_{1} ≤ c_{1}∥f∥_{1}ε^{1−α}.

Where c_{1} = \frac{10a}{α(1−α)}. Proof. We assume that f ∈ C_{∗} and ∫ f = 1. First, observe that

∥P^{n_{ε}}f − ℙ_{ε}f∥_{1} ≤ ∥f − 𝔸_{ε}f∥_{1}.

Next, we recall the estimates

\begin{matrix}f(x) ≤ & ax^{−α} \\ \frac{f(y)}{f(x)} ≤ & (\frac{x}{y})^{1+α} ∀x ≥ y. \end{matrix}

This allows us to bound the L^{1} norm of the difference between the function f and its average.

∥f − 𝔸_{ε}f∥_{1} ≤\frac{1}{2ε} ∫_{ε}^{1−ε} dx ∫_{B_{ε}(x)} dy |f(y) − f(x)| + ∫_{B_{ε}(0)} dx |f(x) − \frac{1}{2ε} ∫_{B_{ε}(x)} dy f(y)|
≤\frac{1}{2ε} ∫_{ε}^{1−ε} dx \{∫_{x−ε}^{x} dy[f(y) − f(x)] + ∫_{x}^{x+ε} dy[f(x) − f(y)]\}

+ ∫_{B_{ε}(0)} f(x)dx + ∫_{B_{2ε}(0)} f(y)dy

≤\frac{1}{2ε} ∫_{ε}^{1−ε} dx ∫_{x−ε}^{x} dyf(y) [1 − \frac{f(y + ε)}{f(y)}] + 4a ∫_{0}^{2ε} x^{−α}dx
≤\frac{a}{2(1 − α)ε} ∫_{ε}^{1−ε} dx (2x^{1−α} − (x + ε)^{1−α} − (x − ε)^{1−α})

+ \frac{a}{2α} ∫_{ε}^{1−ε} dx (x^{−α} − (x + ε)^{−α}) + \frac{4a}{1 − α}(2ε)^{1−α}

≤\frac{a}{2(1 − α)(2 − α)ε} \{(2ε)^{2−α} − 2ε^{2−α} − (1 − 2ε)^{2−α} − 1 + 2(1 − ε)^{2−α}\}

+ \frac{a}{2α(1 − α)}\{2^{1−α}ε^{1−α} − 1 − ε^{1−α} + (1 − ε)^{1−α}\} + \frac{8a}{1 − α}ε^{1−α}

≤\frac{10}{α(1 − α)}ε^{1−α}.

This proves Lemma 3.1. □

Next,

\begin{matrix}ℙ_{ε}f(x) = & ∑_{y∈T^{−nε} x} \frac{1}{2εD_{y}T^{n_{ε}}} ∫_{0}^{1} dzχ_{B_{ε}(y)}(z)f(z) \\ = & ∑_{y∈T^{−nε} x} \frac{1}{2εD_{y}T^{n_{ε}}} ∫_{0}^{1} dzχ_{B_{ε}(z)}(y)f(z) \\ = & \frac{1}{2ε} ∫_{0}^{1} dzP^{n_{ε}}χ_{B_{ε}(z)}(x)f(z) \\ := & ∫_{0}^{1} K_{ε}(x, z)f(z)dz. \end{matrix}

Our task is to find a lower bound for the kernel K_{ε}(x, z). For this purpose, let us define T_{1} to be the map T restricted to the interval [0, 1/2] and a_{n} = T_{1}^{−n}1. We have the following asymptotic bound for the sequence a_{n}.

Lemma 3.2. For all integer n > 0 the following holds

a_{n} ≤ 2^{\frac{1}{α^{2}} + \frac{1}{α}} n^{− \frac{1}{α}} .

Proof. The Lemma is proven by induction. First it is clearly satisfied for n = 1. Next, let us suppose that a_{n} < cn^{− \frac{1}{α}}, and let us prove that a_{n+1} < c(n + 1)^{− \frac{1}{α}}. If it is false, then

a_{n} = a_{n+1}(1 + 2^{α}a_{n+1}^{α}) ≥ c(n + 1)^{− \frac{1}{α}} (1 + 2^{α}c^{α}(n + 1)^{−1}).

By the assumption on a_{n} we obtain

n^{− \frac{1}{α}} ≥ (n + 1)^{− \frac{1}{α}} (1 + \frac{2^{α}c^{α}}{n + 1})

or equivalently

(1 + \frac{1}{n})^{\frac{1}{α}} ≥ 1 + \frac{2^{α}c^{α}}{n + 1}.

By convexity it follows

(2^{\frac{1}{α}} − 1)\frac{1}{n} ≥ \frac{2^{α}c^{α}}{n + 1},

that is

c^{α} ≤ 2^{−α}(2^{\frac{1}{α}} − 1)\frac{n + 1}{n} ≤ 2^{−α+1}(2^{\frac{1}{α}} − 1),

which is contradictory if we choose c = 2^{\frac{1}{α^{2}} + \frac{1}{α}}. □

We define Δ_{k} = [a_{k}, a_{k−1}] for each k > 0. We are now able to prove Proposition 3.3. There exists γ > 0 such that for each ε > 0, x, z ∈ S^{1}

K_{ε}(x, z) ≥ γ,

provided we choose n_{ε} = [2^{2+ \frac{1}{α}} ε^{−α}] + 1.^{5}

Proof. First of all, we choose k_{0} = 3. Next, notice that for each interval J and integer m

(P^{m}χ_{J})(x) ≥ χ_{T^{m}J}(x) inf_{y∈J}(D_{y}T^{m})^{−1}.

Let δ_{0} = a_{k_{0}} − a_{k_{0}+1}. By Lemma 2.4 it is obvious that there exists n_{0} and c_{0} such that for all intervals I of size larger than δ_{0} holds

P^{n}χ_{I} ≥ c_{0}

provided n ≥ n_{0}. Thus the task is to control the inf_{y∈J}(D_{y}T^{m})^{−1}, where m is the time needed for the interval J to become an interval of size δ_{0}. Let I_{0} = [0, a_{k_{0}}]. Let J be an interval, three possibilities can occur :

(1) J ∩ I_{0} = ∅.

(2) J ∩ I_{0} ≠ ∅ and J contains, at most, one a_{k} for k > k_{0}.

(3) J contains more than one a_{k} for k > k_{0}. We can associate to each J a sequence n_{1}, k_{1}, …, k_{p−1}, n_{p} of integers (n_{1} may be null), retracing the trajectory of J in the following way : For a time n_{1} (1) holds, then the image of J enters the intermittent region I_{0} and (2) holds with k = k_{1} +k_{0}, so after k_{1} iterations it exits from I_{0}. Then the image of J stays in the hyperbolic region for n_{2} iterations, and so on... Finally we end when the size of the interval becomes larger than δ_{0} or if case (3) happens. Let us see what happens in this regimes.

(1) Let D = sup_{y∈[a_{k_{0}+1},1]} \frac{D_{y}^{2}T}{D_{y}T^{2}} and r = (D_{a_{k_{0}+1}}T)^{−1} < 1. For n ≤ n_{1}, the usual

distortion estimates yield, for each y ∈ J, with c_{2} = D/(1 − r),

\frac{D_{y}T^{n}|J|}{|T^{n}J|} ≤ Exp[c_{2}|T^{n}J|].

(2) Let J_{1} = T^{n_{1}}J. Let us see what happens in the intermittent region. Suppose that J_{1} ⊂]a_{k+1}, a_{k−1}[, i.e. (2) occurs with k = k_{0} + k_{1}. In this case a direct computation for j ≤ k_{1} and y_{1}, y_{2} ∈ J_{1} implies

\frac{D_{y_{1}}T^{j}}{D_{y_{2}}T^{j}} ≤Exp [∑_{i=1}^{k_{1}} \frac{sup_{ξ∈[a_{k−i+2},a_{k−i}]} D_{ξ}^{2}T}{inf_{ξ∈[a_{k−i+2},a_{k−i}]} D_{ξ}T} |T^{i}y_{2} − T^{i}y_{1}|]

≤Exp [∑_{i=1}^{k_{1}} α(1 + α)a_{k−i+2}^{α−1}(D_{a_{k+2−i}}T^{k_{1}−i})^{−1}|T^{k_{1}}J_{1}|]

≤Exp [α(α + 1) ∑_{q=k_{0}+2}^{k_{0}+k_{1}+1} a_{q}^{α−1}(D_{a_{q}}T^{q−(k_{0}+2)})^{−1}|T^{k_{1}}J_{1}|] .

5Here the square braket stands for the integer part.

Since the first branch of the map is convex, we have

D_{a_{q}}T^{q−(k_{0}+2)} ≥ r \frac{a_{k_{0}+1} − a_{k_{0}+2}}{a_{q−1} − a_{q}} = ra_{k_{0}+2}^{1+α}a_{q}^{−(1+α)}.

Moreover, Lemma 3.2 gives a_{q}^{2α} ≤ 2^{2+2/α}q^{−2}. This yields, setting c_{3} =

\frac{α(1 + α)2^{2+2/α}}{ra_{k_{0}+2}^{1+α}},

\frac{D_{y_{1}}T^{j}}{D_{y_{2}}T^{j}} ≤ Exp [c_{3}|T^{k_{1}}J_{1}|] .

(3) Finally, let us see what happens if (3) holds for some iterate K = T^{j}J. If more than one third of the size of K is in [a_{1}, 1], then we consider K ∩[a_{1}, 1] and case (1) will hold for ever, loosing just factor 1/3. Else we cut K into pieces Δ_{k_{−}}, ⋯, Δ_{k_{+}} such that the union of them is of size bigger than |K|/3. For these Δ_{k}, the preceding computation yields

P^{k−k_{0}}χ_{Δ_{k}} ≥ χ_{Δ_{k_{0}}} \frac{Exp[−c_{3}|Δ_{k_{0}}|]}{|Δ_{k_{0}}|}|Δ_{k}|.

Therefore, with l = n_{0} + k_{+} − k_{0},

P^{l}χ_{K} ≥ ∑_{k=k_{−}}^{k_{+}} P^{l+k_{0}−k}P^{k−k_{0}}χ_{Δ_{k}} ≥ ∑_{k=k_{−}}^{k_{+}} c_{0} \frac{Exp[−c_{3}δ_{0}]}{δ_{0}}|Δ_{k}| ≥ c_{0} \frac{Exp[−c_{3}δ_{0}]}{δ_{0}} \frac{|K|}{3}.

Since we control what happens on each region, it is possible to estimate the total distortion after m = n_{1} + k_{1} + ⋯ + n_{p} + l iterations, where l = n_{0} if case (3) never happens (l = n_{0} + k_{+} − k_{0} if case (3) occurs).

P^{m}χ_{J} ≥P^{l}P^{n_{p}}P^{k_{p−1}} ⋯ P^{n_{2}}P^{k_{1}}P^{n_{1}}χ_{J}
≥|J|\frac{c_{0}}{3δ_{0}} Exp [−c_{3}δ_{0} − c_{2}|T^{n_{p}+···+k_{1}+n_{1}}J| − ⋯ − c_{3}|T^{k_{1}+n_{1}}J| − c_{2}|T^{n_{1}}J|)]
≥|J|\frac{c_{0}}{3δ_{0}} Exp [−(c_{2} + c_{3})δ_{0}(1 + r^{n_{p}} + r^{n_{p}+n_{p−1}} + ⋯ + r^{n_{p}+n_{p−1}+···+n_{2}})]
≥|J|\frac{c_{0}}{3δ_{0}} Exp [\frac{−(c_{2} + c_{3})δ_{0}r}{1 − r} ] := γ|J|.

To conclude, we need to fix n_{ε}. We choose the supremum over all possible values of m = n_{1} + k_{1} + ⋯ + n_{p} + l, associated to intervals J of size 2ε. It is immediate to see that the worst case scenario is when case (3) happens at the beginning, and J =] − 2ε/3, 4ε/3[. In this case, m is such that a_{k_{0}+m} ≤ 2ε/3. Clearly, n_{ε} = [2^{2+ \frac{1}{α}} ε^{−α}] + 1 is large enough and the Lemma is proven. □

§4 Decay of Correlations.

Proposition 3.3 allows immediately to conclude that ℙ_{ε} has an invariant density h_{ε} to which it converges exponentially fast^{6} in L^{1}. In the following we will call μ the invariant measure dμ = hdx. Using all the above facts, we are able to prove our main result.

Theorem 4.1. For all g ∈ L^{∞}, f ∈ C^{(1)}([0, 1]) such that ∫ fdμ = 0 the following holds

|∫ g ∘ T^{n}fdμ| ≤ c_{4}C (∥f∥_{C^{(1)}}) ∥g∥_{∞}n^{− \frac{1}{α} +1}(log n)^{\frac{1}{α}} .

Where C : IR → IR is affine.

Proof. Let f ∈ C_{∗} + IR, ∫_{0}^{1} f = 0 and g ∈ L^{∞}, ∥g∥_{∞} = 1. For each n ∈ ℕ let us write n = kn_{ε} + m, k ∈ ℕ and m < n_{ε}. Remembering Lemma 3.1, we have

|∫_{0}^{1} gP^{n}f| ≤∥P^{n}f − ℙ_{ε}^{k}P^{m}f∥_{1} + ∥ℙ_{ε}^{k}P^{m}f∥_{1}

≤ ∑_{i=0}^{k−1} ∥P^{(i+1)n_{ε}}P^{m}f − ℙ_{ε}P^{in_{ε}}P^{m}f∥_{1} + Exp [−γk] ∥f∥_{1}

≤2c_{1}∥f∥_{1} \frac{n}{n_{ε}} ε^{1−α} + e^{γ}Exp [−γ \frac{n}{n_{ε}} ] ∥f∥_{1}
≤c_{4}∥f∥_{1}n^{− \frac{1}{α} +1}(log n)^{\frac{1}{α}} .

by choosing ε = n^{− \frac{1}{α}} (log n^{(1−γ)^{−1}( \frac{1}{α} −1)2^{2+ \frac{1}{α}}})^{\frac{1}{α}}.

This is not yet the decay of correlation with respect to the absolutely continuous invariant measure dμ = hdx of our dynamical system. To get such a result, we need to notice that if f ∈ C^{(1)}, then we can choose λ, ν, δ ∈ IR such that f_{λ,ν,δ}(x) := (f(x) + λx + ν)h(x) + δ ∈ C_{∗}, and (λx + ν)h(x) + δ ∈ C_{∗}, the dependence of the parameters with respect to the C^{(1)} norm of f being affine. Finally, the decay of correlations with respect to μ, for each f ∈ C^{(1)}, ∫ fdμ = 0 and g ∈ L^{∞} can be estimated as follows

|∫ g ∘ T^{n}fdμ| ≤ c_{4}C (∥f∥_{C^{(1)}}) n^{− \frac{1}{α} +1}(log n)^{\frac{1}{α}} .

Where C : IR → IR is affine. □

≤ 2 ∫_{Ω_{+}} dyf(y) ∫_{Ω} dx[K_{ε}(x, y) − γ] = (1 − γ)2 ∫_{Ω_{+}} dyf(y)
= (1 − γ)∥f∥_{1}.
∥(ℙ_{ε} − Π)^{n}f∥_{1} = ∥ℙ_{ε}^{n}(11 − Π)f∥_{1} ≤ (1 − γ)^{n}∥f∥_{1}.

6Let us briefly recall the argument: set Ω = [0, 1] and consider f ∈ L^{1}(Ω) with ∫_{Ω} f = 0.

Remember that ℙ_{ε}1 = ℙ_{ε}^{∗}1 = 1 and let Ω_{ε}^{+} = {x ∈ Ω | ℙ_{ε}f ≥ 0}; Ω_{+} = {x ∈ Ω | f ≥ 0}, then

∥ℙ_{ε}f∥_{1} = 2 ∫_{Ω_{ε}^{+}} dx ∫_{Ω} dyK_{ε}(x, y)f(y) = 2 ∫_{Ω} dyf(y) ∫_{Ω_{ε}^{+}} dx[K_{ε}(x, y) − γ] Accordingly, for each f ∈ L^{1}(Ω) and defining Πf = ∫_{Ω} f, holds

Remark 4.2.

(1) The proof of Theorem 4.1 allows to estimate the difference between h_{ε} and h; namely, to prove the estimate:^{7} ∥h_{ε} − h∥_{1} ≤ const.ε^{1−α}(log ε^{−1})^{\frac{1}{α}}.

(2) Theorem 4.1 allows to get an estimate for the rate of decay for Hölder continuous observables. More precisely, given a β-Hölder continuous function f, we can bound the correlations by n^{−β( \frac{1}{α} −1)}, up to some logarithmic correction.^{8}

(3) We have obtained a polynomial decay, with a bound comparable to the one found in [Mo] for the piecewise linear case and the one stated in [Is] for the general case. Moreover, compared with numerical simulations [LSV], our bound appears to be extremely close to optimal: the expected one is the same apart from the logarithmic correction.

(4) As a side consequence of our work we have that ∑_{n=0}^{∞} P^{n}f converges in L^{1} provided α < \frac{1}{2}. According to [Li3] this estimate suffices to prove the Central Limit Theorem. That is, given f ∈ C^{(1)}([0, 1]) such that ∫_{0}^{1} fdμ = 0

\frac{1}{\sqrt{n}} ∑_{i=0}^{n} f ∘ T^{i}

converges, in distribution, to a Gaussian variable with variance ∫_{0}^{1} f^{2}dμ + 2 ∑_{i=1}^{∞} ∫_{0}^{1} ff ∘ T^{i}dμ.

§5 General Considerations.

The reader may be under the impression that the proposed approach is specific to the special maps studied here. In particular, section two seems quite model dependent. In fact, while the estimates done there would apply certainly to similar maps, it is true that some more work is needed to present them in a completely general fashion^{9}.

Nevertheless, section two is not completely necessary. Its aim was to make the paper self contained and to emphasize the existence of a very direct method of

∥h − h_{ε}∥_{1} ≤ ∥P^{n}1 − h∥_{1} + ∥ℙε^{\frac{n}{nε}} 1 − h_{ε}∥_{1} + ∥ℙε^{\frac{n}{nε}} 1 − P^{n}1∥_{1}
≤ const.\{n^{1− \frac{1}{α}} (log n)^{\frac{1}{α}} + (1 − γ)^{nε^{α}} + nε\}

7It suffices to write and to choose n ∼ ε^{−α} log ε^{−1}.

8The result is easily obtained by approximation. If f is a β-Hölder function, then we can approximate it in L^{1} by functions f_{ε} ∈ C^{(1)} such that ∥f −f_{ε}∥_{1} ≤ ∥f∥_{β}ε^{β} and ∥f′∥_{∞} ≤ ∥f∥_{β}ε^{β−1} (where ∥f∥_{β} is the β-Hölder norm). Accordingly,

|∫ f ∘ T^{n}g| ≤ ∥g∥_{∞}∥f − f_{ε}∥_{1} + (c_{1}∥f′_{ε}∥_{∞} + c_{2}∥f∥_{∞})n^{− \frac{1}{α} +1}, and the result follows by choosing ε judiciosly.

9Yet, the results extend immediately to any map of the interval which is C^{(1)}–conjugate to our model. More precisely, suppose that T̃ : [0, 1] → [0, 1] and Φ ∈ Diff^{(1)}([0, 1]) satisfies T̃∘Φ = Φ∘T.

Then, for the absolutely continuous T̃–invariant measure μ̃ defined by μ̃(f) = μ(f ∘ Φ), it is straightforward to see that the power law decay is the same for T and T̃ for C^{(1)} observable.

studying the invariant measure. If one is willing to make same assumptions on the invariant measure (which can eventually be proven separately) sections two can be greatly simplified.

Here we discuss briefly what is really essential in order to apply the present method.

Let us consider a map T : [0, 1] → [0, 1] expanding but for the fixed point 0. where D_{0}T = 1. Assume D^{2}T continuous but in the point 0 and

(5.1)  \begin{aligned}|D_{x}^{2}T | & ≤ Cx^{α−1} \\ |D_{x}T | & = 1 + cx^{α} + o(x^{α}). \end{aligned}

Assume further that there exists an invariant probability measure absolutely continuous with respect to Lebesgue (see the discussion at the end of this section). Calling h its density we have P h = h. Then, in analogy with Lemma 2.1 holds^{10}

Lemma 5.1. There exist a > α and b > 0 such that the cone

C = \{f ∈ C^{(1)}(]0, 1]) | 0 ≤ f(x) ≤ 2h(x)∫_{0}^{1}f; |f′(x)| ≤ \frac{a + bx}{x}f(x)\} ,

is invariant with respect to the Perron-Frobenius operator.

Proof. Obviously, the first condition is invariant by P. If f belongs to the cone

|(P f)′(x)| = | ∑_{y∈T^{−1}x} \frac{D_{y}^{2}T}{|D_{y}T |^{3}} f(y) + \frac{1}{|D_{y}T |^{2}} f′(y)|

≤ ∑_{y∈T^{−1}x} (\frac{2Cy^{α−1}}{|D_{y}T |^{2}} + \frac{a + by}{y|D_{y}T |}) \frac{f(y)}{|D_{y}T |}

≤ \frac{a + bx}{x}P f(x) sup_{y∈[0,1]} [\frac{2C}{|D_{y}T |^{2}} \frac{y^{α−1}T (y)}{a + bT (y)} + \frac{T (y)}{y|D_{y}T |} \frac{a + by}{a + bT (y)}]

Let Ω(y) being the term in brackets. We have

T (y) = ∫_{0}^{y} D_{t}T dt = ∫_{0}^{y}(1 + ct^{α} + o(t^{α}))dt = y + \frac{c}{α + 1}y^{α+1} + o(y^{α+1}).

Which shows that Ω(y) ≤ 1 + (\frac{2C}{a} + c(\frac{1}{α+1} − 1))y^{α} + o(y^{α}). Therefore there exists δ > 0 such that Ω(y) ≤ 1 on [0, δ] provided a is big enough. Next, outside this neighborhood, we have y > δ and |D_{y}T | > γ > 1, so

Ω(y) ≤ \frac{1}{a} \frac{2Cδ^{α−1}}{γ^{2}} + \frac{a}{b} \frac{1}{δγ} + \frac{1}{γ} < 1

\begin{aligned}|D_{x}^{2}T | & ≤ Cθ(x)^{α−1} \\ |D_{x}T | & = 1 + c_{j}θ(x)^{α} + o(θ(x)^{α}). \end{aligned}

10Such a result should extends easily to maps expanding but for some fixed points {p_{j}} where D_{p_{j}} T = 1. We can define θ(x) = min_{j} |x − p_{j}| and assume D^{2}T continuous but in the points {p_{j}} and Then the same cone with \frac{a+bθ(x)}{θ(x)} instead of \frac{a+bx}{x} should be invariant.

A PROBABILISTIC APPROACH TO INTERMITTENCY 13

provided a and b/a are big enough. □

In addition, we assume that for some 0 < β < 1, 0 < γ_{0} < 1 and c_{∗} > 0,

(5.2)  γ_{0} ≤ h(x) ≤ c_{∗}x^{−β}.

It follows that a sharper cone is invariant. Note that in the process of proving the next lemma we will establish the analogous of Lemma 2.4. Lemma 5.2. There exists δ > 0 and 0 < γ < γ_{0} such that

C_{∗} = {f ∈ C | f(x) ≥ γ ∫_{0}^{1} f for x ≤ δ},

is invariant with respect to the Perron-Frobenius operator. Proof. The proof starts by establishing the analogous of Lemma 2.4 Sub-Lemma 5.3. If f ∈ C, then

min_{[δ,1]} f ≥ \frac{δ^{a}}{2e^{b}} ∫_{0}^{1} f,

provided δ is chosen small enough. Proof. We have the bounds

f(x) ≤ 2c_{∗}x^{−β} ∫_{0}^{1} f

and

|f′(x)| ≤ (ax^{−1} + b)f(x)

coming from the cone and the assumed estimate on the invariant measure. For x ≥ y > δ the second bound yields

(\frac{y}{x})^{a} e^{−b(x−y)} ≤ \frac{f(x)}{f(y)} ≤ (\frac{x}{y})^{a} e^{b(x−y)}.

Accordingly, normalizing ∫ f = 1,

1 = ∫ f = ∫_{0}^{δ} f + ∫_{δ}^{1} f ≤ \frac{2c_{∗}}{1 − β} δ^{1−β} + δ^{−a}e^{b} min_{x∈[δ, 1]} f.

Next, by choosing δ sufficiently small, we have \frac{2c_{∗}}{1−β} δ^{1−β} ≤ \frac{1}{2} from which the result follows. □

Let μ = ∥DT ∥_{∞}. We choose δ so small that for all y ≤ δ hold |D_{y}T |^{−1} ≥ 1−2cδ^{α} and Sub-Lemma 5.3 together with 1 − 2cδ^{α} + μ^{−1} > 1. Let x ≤ δ, then T^{−1}x will consist of, at least, two points y_{1} ≤ δ and y_{2} ≥ δ. By choosing γ small, holds

\begin{matrix}P f(x) ≥ & |D_{y_{1}}T |^{−1}f(y_{1}) + μ^{−1}f(y_{2}) ≥ [(1 − 2cδ^{α})γ + μ^{−1} min \{γ, \frac{δ^{a}}{2e^{b}} \}] ∫_{0}^{1} P f \\ ≥ γ ∫_{0}^{1} P f. \end{matrix}

□

This is enough to show that inf f ≥ γ ∫ f whenever f ∈ C_{∗}, which implies (since the constant function 1 belongs to the cone C_{∗})

(5.3)  inf_{n≥0} inf_{[0,1]} P^{n}1 ≥ γ > 0.

We have so generalized the two ingredients of section two used in the paper: the existence of an invariant cone for P that allows to obtain the estimates used to prove Lemma 3.1 and the analogous of Lemma 2.4, necessary in proving Proposition 3.3.

The reader can then easily generalize section three, since the distortion estimates depend only on the behavior of the neutral fixed point which is ensured by our assumption on D^{2}T. Section four follows exactly in the same way yielding a polynomial decay depending on α and β.

In conclusion, the following more general result holds:

Proposition 5.4. Given a map T satisfying (5.1), if it has an absolutely continuous invariant probability measure with density satisfying (5.2), then for all g ∈ L^{∞}, f ∈ C^{(1)}, ∫ f = 0, 0 < γ < \frac{1}{α} − \frac{β}{α}, holds

|∫ fg ∘ T^{u}| ≤ c_{4}C(∥f∥_{C^{(1)}})∥g∥_{∞}n^{−γ}

for some constants c_{4} and C(∥f∥_{C^{(1)}}).

Here we do not address how to obtain the needed estimate on the invariant density, although several approaches (besides the ones in the style of what we have done in section two) are possible (see, for example, [Th] where a result of the type (5.2) is obtained by inducing).

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Liverani Carlangelo, Mathematics Department, University of Rome II, Tor Vergata, 00133 Rome, Italy.

Saussol Benoît and Vaienti Sandro, Centre de Physique Théorique, CNRS, Luminy case 907, 13288 Marseille Cedex 9, France. PhyMat, Mathematics Department, University of Toulon, BP 132, 83957 La Garde Cedex, France.

   E-mail address: liverani@mat.utovrm.it, saussol@cptsg3.univ-mrs.fr, vaienti@cptsg2.univ-
mrs.fr