Linear response in the intermittent family: differentiation in a weighted $C^0$-norm

We provide a general framework to study differentiability of SRB measures for one dimensional non-uniformly expanding maps. Our technique is based on inducing the non-uniformly expanding system to a uniformly expanding one, and on showing how the linear response formula of the non-uniformly expanding system is inherited from the linear response formula of the induced one. We apply this general technique to interval maps with a neutral fixed point (Pomeau-Manneville maps) to prove differentiability of the corresponding SRB measure. Our work covers systems that admit a finite SRB measure and it also covers systems that admit an infinite SRB measure. In particular, we obtain a linear response formula for both finite and infinite SRB measures. To the best of our knowledge, this is the first work that contains a linear response result for infinite measure preserving systems.


1.
Introduction. In physical applications of dynamical systems, it is important to understand how statistical properties of a perturbed physical system are related to statistical properties of the original system; i.e., before the occurrence of the perturbation. In particular, it is always desirable to write a first order approximation of the Sinai-Ruelle-Bowen (SRB) measure of the perturbed system in terms of the SRB measure of the original system. In smooth ergodic theory, this direction of research, which was pioneered by David Ruelle, is called differentiation (with respect to noise) of SRB measures. In the physics literature the equivalent term is called 'linear response'.
Linear response has been proved for several classes of smooth dynamical systems that admit exponential, or at least summable, decay of correlations [4,6,8,9,10,11,16]. Negative results, where linear response does not hold, are also known [4,5]. A recent survey on the progress in this area of research is [5]. More recently, results on the linear response of polynomially mixing systems that admit a probabilistic SRB measure were announced in [7,12]. Such systems have attracted the attention of both mathematicians [14,17] and physicists because of their importance in the study of intermittent transition to turbulence [15].
In this work we provide a general framework to study differentiability of SRB measures for one dimensional non-uniformly expanding maps. We use this general framework to study linear response of maps with neutral fixed points. In particular, we apply our results to study linear response of Pomeau-Manneville type maps [14,15]. The difference between our result and those of [7,12] is two-fold: in [7,12] the authors obtain results only for probabilistic SRB measures. Moreover, they obtain a weak form of differentiability. While in our work, we cover both the finite and infinite SRB measure cases and we prove differentiability in norm 1 . Moreover, we provide a linear response formula that covers both the finite and infinite SRB measure cases.
In Section 2 we introduce a general setup for the systems we study and we state our assumptions on this general setup. Section 3 includes the statement of our main results (Theorems 3.1 and 3.2). Section 4 contains the proof of the theorems through several lemmas. In Section 5 we show that the assumptions of Section 2 are satisfied by the intermittent maps studied in [14].

2.1.
Interval maps with an inducing scheme. We introduce now a class of (family of) interval maps which are non-uniformly expanding with two branches, for which one can construct an inducing scheme which allow to inherit the linear response formula from the one for the induced system.
• Let V be a neighbourhood of 0. For any ε ∈ V , T ε :  1]. The inverse branches of T 0,ε , T 1,ε are respectively denoted by g 0,ε and g 1,ε . We call T 0 := T the unperturbed map, and T ε , for ε = 0, the perturbed map. • We assume that for each i = 0, 1 and j = 0, 1, 2 the following partial derivatives exist and satisfy the commutation relation • We assume that T ε has a unique absolutely continuous invariant measure 2 (up to multiplication) whose Radom-Nykodim derivative will be denoted by h ε , and we denote for simplicity h = h 0 . • LetT ε , be the first return map of T ε to ∆, where ∆ := [1/2, 1]; i.e., for x ∈ ∆ where R ε (x) = inf{n ≥ 1 : T n ε (x) ∈ ∆}. We assume thatT ε has a unique acim (up to multiplication) with a continuous density denotedĥ ε ∈ C 0 . • Let Ω be the set of finite sequences of the form ω = 10 n , for n ∈ N ∪ {0}. We set g ω,ε = g 1,ε • g n 0,ε . Then for x ∈ [0, 1] we have T n+1 and where B denotes the set of continuous functions on (0, 1] with the norm for a fixed 3 γ > 0. When equipped with the norm · B , B is a Banach space.
Note that F ε is a linear operator. In fact, for x ∈ [0, 1] \ ∆, the formula of F ε can be re-written using the Perron-Frobenius operator of T ε : where L ε is the Perron-Frobenius operator associated with T ε ; i.e., for ϕ ∈ L ∞ and ψ ∈ L 1 It is well known, see for instance [3], that the densities of the original system and the induced one are related (modulo normalization in the finite measure case) by We also define the following operator, which will represent ∂ ε F ε Φ| ε=0 where a ω = ∂ ε g ω,ε | ε=0 and b ω = ∂ ε g ω,ε | ε=0 .

2.2.
Interval maps with countable number of branches. We introduce here a class of (family of) interval maps which are uniformly expanding, with a finite or countable number of branches, for which we will be able to prove a linear response formula. The induced map in Subsection 2.1 is a particular case of such uniformly expanding maps. Let ∆ be an interval and V be a neighborhood of 0. Let Ω be a finite or countable set. We assume that the mapsT ε : ∆ → ∆ satisfy • For each ε ∈ V , there exists a partition (mod 0) of ∆ into open intervals ∆ ω,ε , ω ∈ Ω such that the restriction ofT ε to ∆ ω,ε is piecewise C 3 , onto and uniformly expanding in the sense that inf ω inf ∆ω,ε |T ω,ε | > 1. We denote by g ω,ε the inverse branches ofT ε on ∆ ω,ε . • We assume that for each ω ∈ Ω and j = 0, 1, 2 the following partial derivatives exist and satisfy the commutation relation 4 • We assume and and for i = 1, 2 LetL ε denote the Perron-Frobenius operator of the mapT ε ; i.e., for Φ ∈ L 1 (∆) for a.e. x ∈ ∆. Under these conditions it is well known thatT ε admits a unique (up to multiplication) invariant absolutely continuous finite measure. We denote its density byĥ ε . HenceL εĥε =ĥ ε . Moreover,L ε has a spectral gap when acting on C k , k = 1, 2 (see for instance [13]). We denote the Perron-Frobenius operator of the unperturbed mapT byL; i.e.,L :=L 0 and letĥ :=ĥ 0 .
3. Statement of the main results.
3.1. Statement of the main results. A first general statement is that the differentiability of the T ε absolutely continuous measure is inherited from that of the induced system.
Theorem 3.1. Let T ε be a family of maps of the interval as described in Subsection 2.1. If the densityĥ ε of the induced mapT ε is differentiable as a C 0 element, that is there existsĥ * ∈ C 0 such that for someĥ ∈ C 0 , then a) there exists h * ∈ B such that i.e., h ε is differentiable as an element of B with respect to ε; b) in particular, if the conditions hold for some γ < 1 c) The function h * is given by 5 Next, we show that for the family of maps with countable number of branches introduced in Subsection 2.2 the invariant density is differentiable as an element of C 0 .
Theorem 3.2. LetT ε : ∆ → ∆ be a family of maps of the interval as described in Subsection 2.2. Then the densityĥ ε of the mapT ε is differentiable as a C 0 element, that is there existsĥ * ∈ C 0 such that (13) holds. Moreover, we have the linear response formulaĥ whereĥ is the spatial derivative ofĥ and Corollary 3.3. If T ε satisfies the assumptions of Subsections 2.1 and 2.2, then Proof. The proof follows from Theorems 3.1 and 3.2.
Remark 3.4 (Moving inducing sets). We notice that Theorem 3.1 generalizes easily to the case where the inducing sets ∆ ε are allowed to depend on ε in a C 1 way. Indeed, any C 1 family of C 1 diffeomorphism S ε : [0, 1] → [0, 1] such that S ε (∆ ε ) = ∆, S 0 = id, will conjugate T ε with a mapT ε whose inducing set is ∆. Applying Theorem 3.1 to the mapT ε , with the obvious notation, we obtain: Then using (15) and the fact that h ε =h ε • S ε · S ε we obtain 5 Note that in the finite measure case, h * is the derivative of the non-normalized density hε. The advantage in working with hε is reflected in keeping the operator Fε linear and to accommodate the infinite measure preserving case. In the finite measure case, once the derivative of hε is obtained, the derivative of the normalized density can be easily computed. Indeed, hε = h + εh * + o(ε).

3.2.
Rigorous numerical approximation of the derivative. An important feature of our approach is that it could be amenable to obtain rigorous numerical approximation of h * . In particular, sinceL has a spectral gap on C k , k = 1, 2, using ideas of [2] one can approximate (I −L) −1L [A 0ĥ + B 0ĥ ] as a first step, and in the second step one can follow the path of [1] and pull back the computed formula of the first step to the full system and obtain a numerical approximation of h * in B.

4.
Proof of the results. We use the letter C to denote positive constants whose values may change when estimating various expressions but are independent of both ε and ω (or n). In the following, we first present in Subsection 4.1 the proof of Theorem 3.2, and then in Subsection 4.2 we present the proof of Theorem 3.1.

Proof of Theorem 3.2.
We first prove a lemma that will be used in the linear response formula in Theorem 3.2.
Proof. One easily checks that SinceL ε has a spectral gap on C 1 it eventually contracts exponentially on the subset of zero average functions C 1 0 . Since the ranges of (L ε −L 0 ) and (I −L ε ) are contained in C 1 0 , the composition below is well defined This completes the proof of the lemma.
Setting H ε =L ε −L and G ε = (I −L ε ) −1 , Lemma 4.2 readŝ We then obtain, using Lemma 4.3 below, the following first order expansion in C 1 We then show, see second statement of Lemma 4.5 below, that G ε is uniformly bounded in L(C 1 0 , C 0 ) to obtain the following expansion in C 0 G ε H εĥ = εG ε q + o(ε).
Finally, using the two expansions above with (19) and showing that G ε (q) → G 0 (q) in C 0 , see the first statement of Lemma 4.5 below, we obtain in C 0 which proves the theorem.
Proof. Recall that H ε =L ε −L hence we need to show that ε →L εĥ is differentiable as a C 1 element, on some neighborhood V of 0. To this end, recall thatL εĥ = ωĥ • g ω,ε g ω,ε . It suffices to show that (i) for each ω, the map ε ∈ V →ĥ • g ω,ε g ω,ε ∈ C 1 is differentiable; We first prove (i). Drop for simplicity the subscript ω and write g ε = g ω,ε and let f ε =ĥ • g ε g ε . We have By the commutation relations given by assumption (9) we have and these are continuous functions on ∆ × V . Let ν ∈ V and ε be small. We have For each x, by the mean value theorem, there exists η i x,ε such that f We conclude by (21) and the commutation relation (20). We now prove (ii).
Lemma 4.5. We have G ε (q) → G 0 (q) in C 0 and G ε is uniformly bounded in L(C 1 0 , C 0 ) Proof. We use the fact that the family of operatorsL ε has a uniform spectral gap on C 1 0 , for ε in a neighborhood of 0. Hence, these operators are invertible on this space and we have (1 −L ε ) −1 This proves in particular the second statement. Note that By Lemma 4.4 with Φ = (I −L) −1 (q) and the previous observations this proves the first statement.

4.2.
Proof of Theorem 3.1. We first prove a lemma that will be used in the linear response formula in Theorem 3.1.
Lemma 4.6. For any differentiable function Φ, the function Φ • g ω,ε g ω,ε is differentiable with respect to ε and we have on [0, 1] Proof. The proof follows by differentiating with respect to ε and is similar to (18).
Strategy of the proof of Theorem 3.1. The argument starts from the first order expansion forĥ ε in C 0ĥ Using this, we then obtain, by the second statement of Lemma 4.8 below and relation (7) the following expansion in B Finally, we obtain by Lemma 4.7 below and the first statement of Lemma 4.8 below the first order expansion of h ε in B which finishes the proof of the theorem.
We skip the proof of (i) as, by using (1), it follows similar steps as in the proof of (i) in Lemma 4.3. For (ii), using (24) of Lemma 4.6 we have where we have used the fact thatĥ is C 1 and assumptions (2) and (3). The rest of the proof follows from assumptions (4) and (5).