Statistical aspects of mean field coupled intermittent maps

Abstract We study infinite systems of mean field weakly coupled intermittent maps in the Pomeau–Manneville scenario. We prove that the coupled system admits a unique ‘physical’ stationary state, to which all absolutely continuous states converge. Moreover, we show that suitably regular states converge polynomially.


Introduction
Mean field coupled dynamics can be thought of as a dynamical system with n "particles" with states x 1 , . . ., x n evolving according to an equation of the type Here T is some transformation, ε ∈ R is the strength of coupling and δ x k are the delta functions, so (δ /n is a probability measure describing the "mean state" of the system. As n → ∞, it is natural to consider the evolution of the distribution of particles: if µ is a probability measure describing distribution of particles, then one looks at the operator that maps µ to the distribution of T (x, εµ), where x ∼ µ is random.
In chaotic dynamics, mean field coupled systems have been studied first when T is a perturbation of a uniformly expanding circle map by Keller [5] and followed, among others, by Bálint, Keller, Sélley and Tóth [2], Blank [3], Galatolo [4], Sélley and Tanzi [9].The case when T is a perturbation of an Anosov diffeomorphism has been covered by Bahsoun, Liverani and Sélley [1] (see in particular [1, Section 2.2] for a motivation of such study).See Galatolo [4] for a general framework when the site dynamics admits exponential decay of correlations.The results of [4] also apply to certain mean field coupled random systems.We refer the reader to Tanzi [10] for a recent review on the topic and to [1] for connections with classical and important partial differential equations.
In this work we consider the situation where T is a perturbation of the prototypical chaotic map with non-uniform expansion and polynomial decay of correlations: the intermittent map on the unit interval [0, 1] in the Pomeau-Manneville scenario [8].We restrict to the case when the coupling is weak, i.e. ε is small.
Our results apply to a wide class of intermittent systems satisfying standard assumptions (see Section 2).To keep the introduction simple, here we consider a very concrete example.
Informally, γ h changes the degree of the indifferent point at 0, and ϕ h is responsible for perturbations away from 0.
We restrict to ε ∈ [−ε 0 , ε 0 ] with ε 0 small and to h nonnegative with and let 3) We call L ε the self-consistent transfer operator.Observe that L ε is nonlinear, and that L ε h is the density of the distribution of T εh (x), if x is distributed according to the probability measure with density h.
Remark 1.4.A curious corollary of Theorem 1.1 is that the density of the unique absolutely continuous invariant probability measure for the map x → x(1 + x γ * ) is smooth, namely C ∞ (0, 1] with the bounds (1.4).Our abstract framework covers such a result also for the Liverani-Saussol-Vaienti maps [7].
To the best of our knowledge, this is the first time such a result is written down.At the same time, we are aware of at least two different unwritten prior proofs which achieve similar or stronger results, one by Damien Thomine and the other by Caroline Wormell.
Remark 1.5.Another example to which our results apply is where γ * ∈ (0, 1) and ε ∈ [0, ε 0 ].This is interesting because now each T εh with ε > 0 is uniformly expanding, but the expansion is not uniform in ε.Thus, even for this example, standard operator contraction techniques employed in [4,5] do not apply.
Remark 1.6.Let h ε be as in Theorem 1.1.A natural question is to study the regularity of the map ε → h ε .We expect that it should be differentiable in a suitable topology.
The paper is organised as follows.Theorems 1.1 and 1.2 are corollaries of the general results in Section 2, where we introduce the abstract framework and state the abstract results.The abstract proofs are carried out in Section 3, and in Section 4 we verify that the specific map (1.1) fits the abstract assumptions.

Assumptions and results
We consider a family of maps and h is a probability density on [0, 1].
We require that each such T εh is a full branch increasing map with finitely many branches, i.e. there is a finite partition of the interval (0, 1) into open intervals B k εh , modulo their endpoints, such that each restriction T εh : B k εh → (0, 1) is an increasing bijection.
We assume that each restriction T εh : B k εh → (0, 1) satisfies the following assumptions with the constants independent of ε, h or the branch: There are b 1 , . . ., b r > 0 and χ * ∈ (0, 1] so that for all 1 ≤ ℓ ≤ r, 0 ≤ j ≤ ℓ and each monomial w ℓ,j in the expansion of (w ℓ ) (ℓ−j) , where χ ℓ (x) = min{x ℓ , χ * }. (For example, the expansion of (w εh is not the leftmost branch, then T εh has bounded distortion: with C d > 0. Remark 2.1.Assumption (c) is unusual, but we did not see a way to replace it with something natural.At the same time, it is straightforward to verify and to apply.It plays the role of a distortion bound in C r adapted to an intermittency at 0.
In addition to the above, we assume that the transfer operators corresponding to T εh vary nicely in h.We state this formally in (2.6), after we introduce the required notation.
Define the transfer operators L εh and L ε as in (1.2) and (1.3).

For an integer
Suppose that a 1 , . . ., a r > 0. For 1 ≤ k ≤ r, let (2.4) Take A > 0 and let where C depends only on a 1 and A.
Now and for the rest of the paper we fix a 1 , . . ., a r and A so that D k and D k 1 are non-empty and invariant under L εh .This can be done thanks to the following lemma.
The proof of Lemma 2.3 is postponed to Section 3.
Our main abstract result is the following theorem: Theorem 2.4.There exists ε 0 > 0 such that for every ε ∈ [−ε 0 , ε 0 ]: (a) There exists h ε in D r 1 so that for every probability density h, , where C depends only on C and Ã, ã1 , ã2 .

Proofs
In this section we prove Lemma 2.3 and Theorem 2.4.The latter follows from Lemma 3.3 and Propositions 3.6, 3.8.
Throughout we work with maps T εh as per our assumptions, in particular ε is always assumed to belong to [−ε * , ε * ], and h is always a probability density.
3.1.Invariance of D q , D q 1 and distortion bounds.We start with the proof of Lemma 2.3.Our construction of A, a 1 , . . ., a r allows them to be arbitrarily large, and without mentioning this further, we restrict the choice so that where Dr 1 is the version of D r 1 with A/2, a 1 /2, . . ., a r /2 in place of A, a 1 , . . ., a r .Informally, we require that (1 − γ)x −γ is deep inside D r 1 .Lemma 3.1.There is a choice of a 1 , . . ., a r such that if B ⊂ (0, 1) is a branch of T εh and g ∈ D q with 1 ≤ q ≤ r, then L εh (1 B g) ∈ D q .Proof.To simplify the notation, let T : B → (0, 1) denote the restriction of T εh to B. Then its inverse T −1 is well defined.Let w = 1/T ′ and f = (gw) • T −1 .We have to choose a 1 , . . ., a r show that f ∈ D q independently of g and B.
For illustration, it is helpful to write out a couple of derivatives of f : An observation that f (ℓ) = (u ℓ w) • T −1 , where u 0 = g and u ℓ+1 = (u ℓ w) ′ , generalizes the pattern: Here each W ℓ,j is a linear combination of monomials from the expansion of (w ℓ ) (ℓ−j) .
By (2.2), for each ℓ there is c ℓ > 0, depending only on b 1 , . . ., b ℓ−1 , such that Using this and the triangle inequality, It is immediate that if g ∈ D q and 1 ≤ ℓ < q, then the right hand side of (3.3) is at most a ℓ /χ ℓ •T , which in turn implies that f (ℓ) /f ≤ a ℓ /χ ℓ .A similar argument yields Lip f (q−1) /f ≤ a q /χ q , hence f ∈ D q as required.
Proof of Lemma 2.3.First we show that part (a) follows from Lemma 3.1.Indeed, let a 1 , . . ., a r be as in Lemma 3.1 and suppose that g ∈ D q .Write where the sum is taken over the branches of T εh .Each L εh (1 B g) belongs to D q by Lemma 3.1, and D q is closed under addition.Hence L εh g ∈ D q .
It remains to prove part (b) by choosing a suitable A. Without loss of generality, we restrict to q = 1.
Fix ε, h and denote, to simplify notation, T = T εh and L = L εh .Suppose that g ∈ D 1 with 1 0 g(s) ds = 1 and x 0 g(s) ds ≤ Ax 1−γ for all x.We have to show that if A is sufficiently large, then x 0 (Lg)(s) ds ≤ Ax 1−γ .Suppose that T has branches B 1 , . . ., B N , where B 1 is the leftmost branch.Denote by T k : B k → (0, 1) the corresponding restrictions.Taking the sum over branches, write with some c depending only on c γ and γ, where c ′ > 0 also depends only on c γ and γ.
, where C depends only on a 1 and χ * .Since T k is uniformly expanding with bounded distortion (2.3), Assembling (3.4), (3.5) and (3.6), we have with c ′ , C ′′ > 0 independent of A, ε and h.For each A ≥ C ′′ /c ′ the right hand side above is bounded by Ax 1−γ , as desired.
A useful corollary of Lemma 3.1 is a distortion bound: Lemma 3.2.Let n > 0 and δ > 0. Consider maps T εh k , 1 ≤ k ≤ n with some ε and h k as per our assumptions.Choose and restrict to a single branch for every T εh k , so that all T εh k are invertible and T −1 εh k is well defined.Denote In particular, for every δ > 0 the bounds above are uniform in x ∈ [δ, 1].
Proof.Let P εh k be the transfer operator for T εh k , restricted to the chosen branch: Let g ≡ 1.Clearly, g ∈ D r .By Lemma 3.1, P εh k D r ⊂ D r and thus P n g ∈ D r .On the other hand, P n g = J n , and the desired result follows from the definition of D k .

Fixed point and memory loss.
Further let h ε be a fixed point of L ε as in the following lemma; later we will show that it is unique.

Lemma 3.3. There exists h
The first term on the right is bounded by f −g L 1 because L εf is a contraction in L 1 .By (2.6), so is the second term, up to a multiplicative constant.It follows that L ε is continuous in L 1 .Recall that L ε preserves D r 1 and note that D r 1 is compact in the L 1 topology.By the Schauder fixed point theorem, L ε has a fixed point in D r 1 .
Further we use the rates of memory loss for sequential dynamics from [6]: The constant C 1 depends only on C, and C 2 depends additionally on γ ′ , C ′ γ .
Proof.In the language of [6], the family T εh k defines a nonstationary nonuniformly expanding dynamical system.As a base of "induction" we use the whole interval (0, 1).For a return time of x ∈ (0, 1) corresponding to a sequence T εh k , k ≥ n, we take the minimal j ≥ 1 such that belongs to one of the right branches of T εh j , i.e. not to the leftmost branch.Note that we work with the return time which is not a first return time, unlike in [6], but this is a minor issue that can be solved by extending the space where the dynamics is defined.
It is a direct verification that our assumptions and Lemma 3.2 verify (NU:1-NU:7) in [6] with tail function h(n) = Cn −1/γ with C depending only on γ and c γ .
Further in the language of [6], functions in D 1  1 are densities of probability measures with a uniform tail bound Cn −1/γ+1 , where C depends only on C; a density of a probability measure with tail bound Cn −1/γ+γ ′ /γ with C depending only on C and γ ′ , C ′ γ .In this setup, Theorem 3.4 is a particular case of [6, Theorem 3.8 and Remark 3.9].
Take g = L n 1.Then g ∈ D r 1 by the invariance of D r 1 , and L n ε f − g ≤ δ by construction.
Proposition 3.8.Suppose that f is a probability density on [0, 1].There is Proof.Choose a small δ > 0. Without loss of generality, suppose that f − f L 1 ≤ δ with f ∈ D 1 1 .(The general case is recovered using Lemma 3.7 and replacing f with L n ε f with sufficiently large n.)As in the proof of Proposition 3.6, denote f n = L n ε f and fn = L n ε f , and write f n − fn = A n + B n , where By (2.6) and Theorem 3.4, where C ′ depends only on C and C ′′ = C ′ ∞ j=1 j −1/γ+β/γ ; recall that −1/γ + β/γ < −1, so this sum is finite.

Example: verification of assumptions
Here we verify that the example (1.1) fits the assumptions of Section 2, namely (a), (b), (c), (d) and (2.6).The key statements are Proposition 4.1 and Corollary 4.3.
In this section we use the notation A B for A ≤ CB with C depending only on ε * , and A ∼ B for A B A. Proof.It is immediate that (a), (b) and (d) hold, so we only need to justify (c).Denote γ = γ * + εγ h and observe that, with w and each w ℓ,j as in (2.2), The implied constants depend on ℓ and j but not on ε or h, and (c) follows.
It remains to verify (2.6).The precise expressions for γ h and ϕ h are not too important, we rely on the following their properties: • ϕ h (0) = ϕ ′ h (0) = ϕ h (1) = 0 for each h, so that, informally, ϕ h has no effect on the indifferent fixed point at 0.