Spectral Gap and quantitative statistical stability for systems with contracting fibers and Lorenz like maps

We consider transformations preserving a contracting foliation, such that the associated quotient map satisfies a Lasota-Yorke inequality. We prove that the associated transfer operator, acting on suitable normed spaces, has a spectral gap (on which we have quantitative estimation). As an application we consider Lorenz-like two dimensional maps (piecewise hyperbolic with unbounded contraction and expansion rate): we prove that those systems have a spectral gap and we show a quantitative estimate for their statistical stability. Under deterministic perturbations of the system of size $\delta$, the physical measure varies continuously, with a modulus of continuity $O(\delta \log \delta )$, which is asymptotically optimal for this kind of piecewise smooth maps.


Introduction
The study of the behavior of the transfer operator restricted to a suitable functional space has proven to be a powerful tool for the understanding of the statistical properties of a dynamical system. This approach gave first results in the study of the dynamics of piecewise expanding maps where the involved spaces are made of regular, absolutely continuous measures (see [5], [22], [28] for some introductory text). In recent years the approach was extended to piecewise hyperbolic systems by the use of suitable anisotropic norms (the expanding and contracting directions are managed differently), leading to suitable distribution spaces on which the transfer operator has good spectral properties (see e.g. [7], [6], [10], [17]). From these properties, several limit theorems or stability statements can be deduced. This approach has proven to be successful in non-trivial classes of systems like geodesic flows (see [22], [9]) or billiard maps (ess e.g. [12] [13] where a relatively simple and unified approach to many limit and perturbative results is given for the Lorentz gas). We remark that in these approaches, usually some condition of boundedness of the derivatives or transversality between the map's singular set and the contracting directions is supposed.
In this work, we consider skew product maps preserving a uniformly contracting foliation. We show how it is possible, in a simple way, to define suitable spaces of signed measures (with an anisotropic norm) such that, under small regularity assumptions, the transfer operator associated to the dynamics has a spectral gap (in the sense given in Theorem 6.1). This shows an exponential convergence to 0 in a certain norm for the iteration of a large class of zero average measures by the transfer operator. We remark that in this approach the speed of this convergence can be quantitatively estimated, and depends on the rate of contraction of the stable foliation, the coefficients of the Lasota-Yorke inequality and the rate of convergence to equilibrium of the induced quotient map (see Remark 6.3). We also remark that in our approach we can deal with maps having C 1+α regularity, having unbounded derivatives, and where the singular set is parallel to the contracting direction, as it happen in the Lorenz-like maps we consider in Section 7.
The function spaces we consider are defined by disintegrating signed measures on the phase space along the contracting foliation. The signed measure itself is then seen as a family of measures on the contracting leaves. We can then consider some notion of regularity for this family to define suitable spaces of more or less "regular"measures where to apply our transfer operator. To give an idea of these function spaces (see section 3), in the case of skew product maps of the unit square I × I to itself, the disintegration gives rise to a one dimensional family (a path) of measures defined on the contracting leaves, each leaf is isomorphic to the unit interval I, hence a measure on I × I is seen as a path of measures on I: a path in a metric space. The function spaces are defined by suitable notions of regularity for these paths. In the case I × I for example, the spaces which arise are included in L 1 (I, Lip(I) ′ ) (the space of L 1 functions from the interval to the dual of the space of Lipschitz functions on the interval), imposing some kind of further regularity. We remark that this is a space of distribution valued functions. For simplicity we will only use normed vector spaces of signed measures in this paper, we do not need to consider the completion of the space of signed measure, which would lead to distribution spaces.
The paper is structured as follows: in Section 3 we introduce the functional spaces we consider; in Section 4 we show the basic properties of the transfer operator when applied to these spaces. In particular we see that there is a useful "Perron-Frobenius"-like formula. In Section 5 we see the basic properties of the iteration of the transfer operator on the spaces we consider. In particular we see Lasota-Yorke inequalities and a convergence to equilibrium statement. In Section 6 we use the convergence to equilibrium and the Lasota-Yorke inequalities to prove the spectral gap. In Section 7 we present an application of our construction, showing a spectral gap for 2-dimensional Lorenz-like maps (piecewise C 1+α hyperbolic maps with unbounded expansion and contraction rates). In Section 8 we apply our construction to a class of piecewise C 2 Lorenz-like maps. We prove stronger (bounded variation) regularity results for the iteration of probability measures on that systems, and use this to prove a strong statistical stability statement with respect to deterministic perturbations: we establish a modulus of continuity δ log δ for the variation of the physical measure in weak space (L 1 (I, Lip(I) ′ )) after a "size δ"perturbation. We remark that a qualitative statement, for a class of similar maps was given in [1]. Acknowledgment

Contracting Fiber Maps
Consider Σ = N 1 × N 2 , where N 1 and N 2 are compact and finite dimensional Riemannian manifolds such that diam(N 2 ) = 1, where diam(N 2 ) denotes the diameter of N 2 with respect to its Riemannian metric, d 2 . This is not restrictive but will avoid some multiplicative constants. Denote by m 1 and m 2 the Lebesgue measures on N 1 and N 2 respectively, generated by their corresponding Riemannian volumes, normalized so that m 1 (N 1 ) = m 2 (N 2 ) = 1 and m = m 1 × m 2 . Consider a map F : (Σ, m) −→ (Σ, m), We suppose that F s is contracted: there exists 0 < α < 1 such that for all x ∈ N 1 it holds (3) d 2 (G(x, y 1 ), G(x, y 2 )) ≤ αd 2 (y 1 , y 2 ), f or all y 1 , y 2 ∈ N 2 .
2.0.2. Properties of T and of its associated transfer operator. Suppose that: T1: T is non-singular with respect to m 1 (m 1 (A) = 0 ⇒ m 1 (T −1 (A))) = 0). T2: There exists a disjoint collection of open sets P = {P 1 , · · · , P q } of N 1 , such that m 1 ( q i=1 P i ) = 1 and T i := T | Pi is a diffeomorphism : P i → T i (P i ) ⊆ N 1 , with det DT i (x) = 0 for all x ∈ P i and for all i, where DT i is the Jacobian of T i with respect to the Riemannian metric of N 1 . T3: Let us consider the Perron-Frobenius Operator associated to T , PT 1 . We will now make some assumption on the existence of a suitable functional analytic setting adapted to PT . Let us hence denote the L 1 m1 norm 2 by | · | 1 and suppose that there exists a Banach space (S , | · | s ) such that T3.1: S ⊂ L 1 m1 is PT -invariant, | · | 1 ≤ | · | s and PT : S −→ S is bounded; T3.2: The unit ball of (S , | · | s ) is relatively compact in (L 1 m1 , | · | 1 ); T3.3: (Lasota Yorke inequality) There exists k ∈ N, 0 < β 0 < 1 and C > 0 such that, for all f ∈ S , it holds

T3.4:
Suppose there is an unique ψ x ∈ S with ψ x ≥ 0 and |ψ x | 1 = 1 such that PT (ψ x ) = ψ x , and if ψ ∈ S is another density for a probability measure, then P k The unique operator PT : 2 Notation: In the following we use | · | to indicate the usual absolute value or norms for signed measures on the basis space N 1 . We will use || · || for norms defined for signed measures on Σ. 3 This assumption ensures that from our point of view the system is indecomposable. For piecewise expanding maps e.g., the assumption follows from topological mixing.
2.1. Theorem. If T satisfy T 3.1, ..., T 3.4 then there exist 0 < r < 1 and D > 0 such that for all φ ∈ S with φ dm 1 = 0 and for all n ≥ 0, it holds The following additional property on | · | s will be supposed sometimes in the paper, to obtain spectral gap on L ∞ like spaces.
norm on N 1 ) Iterating the inequality (4) and since it holds | PT (h)| 1 ≤ |h| 1 , for all h ∈ L 1 m1 , we have for all f ∈ S and for all l ∈ N. For a given n ∈ N, set n = q n k + r n , where 0 ≤ r n ≤ k. Since PT : S −→ S is bounded, there exists M 1 > 0 such that 3. Weak and strong spaces 3.1. L 1 like spaces. Through this section we construct some function spaces which are suitable for the systems we consider. The idea is to consider spaces of signed measures, with suitable norms constructed by disintegrating measures along the stable foliation. Thus a signed measure will be seen as a family of measures on each leaf. As an example, a measure on the square will be seen as a one parameter family (a path) of measures on the interval (a stable leaf). In the vertical, contracting direction (on the leaves) we will consider a norm which is the dual of the Lipschitz norm. In the "horizontal"direction we will consider essentially the L 1 norm.
Consider a probability space (Σ, B, µ) and a partition Γ of Σ by measurable sets γ ∈ B. Denote by π : Σ −→ Γ the projection that associates to each point x ∈ M the element of Γ which contains x, i.e. π(x) = γ x . Let B be the σ-algebra of Γ provided by π. Precisely, a subset Q ⊂ Γ is measurable if, and only if, π −1 (Q) ∈ B. We define the quotient measure µ x on Γ by µ x (Q) = µ(π −1 (Q)).
The proof of the following theorem can be found in [24], Theorem 5.1.11.
3.1. Theorem. (Rokhlin's Disintegration Theorem) Suppose that Σ is a complete and separable metric space, Γ is a measurable partition 4 of Σ and µ is a probability on Σ. Then, µ admits a disintegration relatively to Γ, i.e. a family {µ γ } γ∈Γ of probabilities on Σ and a quotient measure µ x = π * µ such that, for all measurable set E ⊂ Σ: The proof of the following lemma can be found in [24], proposition 5.1.7.
Let (X, d) be a compact metric space, g : X −→ R be a Lipschitz function and let L(g) be its best Lipschitz constant, i.e.
3.3. Definition. Given two signed measures µ and ν on X, we define a Wasserstein-Kantorovich Like distance between µ and ν by From now, we denote (9) ||µ|| W := W 0 1 (0, µ). As a matter of fact, || · || W defines a norm on the vector space of signed measures defined on a compact metric space. We remark that this norm is equivalent the dual of the Lipschitz norm.
Let SB(Σ) be the space of Borel signed measures on Σ. Given µ ∈ SB(Σ) denote by µ + and µ − the positive and the negative parts of it (µ = µ + − µ − ). Denote by AB the set of signed measures µ ∈ SB(Σ) such that its associated positive and negative marginal measures, π * x µ + and π * x µ − are absolutely continuous with respect to the volume measure m 1 , i.e.
Given a probability measure µ ∈ AB on Σ, theorem 3.1 describes a disintegration {µ γ } γ , µ x along F s (see equation (2)) 5 by a family {µ γ } γ of probability measures on the stable leaves 6 and, since µ ∈ AB, µ x can be identified with a non negative marginal density φ x : N 1 −→ R, defined almost everywhere, with |φ x | 1 = 1. For a positive measure µ ∈ AB we define its disintegration by disintegrating the normalization of µ.

3.4.
Definition. Let π γ,y : γ −→ N 2 be the restriction π y | γ , where π y : Σ −→ N 2 is the projection defined by π y (x, y) = y and γ ∈ F s . Given a positive measure µ ∈ AB and its disintegration along the stable leaves F s , {µ γ } γ , µ x = φ x m 1 , we define the restriction of µ on γ as the positive measure µ| γ on N 2 (not on the leaf γ) defined, for all mensurable set A ⊂ N 2 , as . For a given signed measure µ ∈ AB and its decomposition µ = µ + − µ − , define the restriction of µ on γ by (12) µ| 3.5. Definition. Let L 1 ⊆ AB be defined as and define a norm on it, || · || 1 : L 1 −→ R, by Now, we define the following set of signed measures on Σ, Consider the function || · || S 1 : S 1 −→ R, defined by x being the marginals of µ ± as explained before. φ x is the marginal density of the disintegration of µ and we remark that φ + x is not necessarily equal to the positive part of φ x . The proof of the next proposition is straightforward. Details can be found in [23].
3.6. Proposition. L 1 , || · || 1 and S 1 , || · || S 1 are normed vector spaces. 5 By lemma 3.2, the disintegration of a measure µ is the µ x -unique measurable family ({µ γ }γ , φ x m 1 ) such that, for every measurable set E ⊂ Σ it holds (11) µ We also remark that, in our context, Γ and π of theorem 3.1 are respectively equal to F s and πx, defined by π(x, y) = x, where x ∈ N 1 and y ∈ N 2 . 6 In the following to simplify notations, when no confusion is possible we will indicate the generic leaf or its coordinate with γ.

Transfer operator associated to F
Let us now consider the transfer operator F * associated with F , i.e. such that for each signed measure µ ∈ SB(Σ) and for each measurable set E ⊂ Σ.
is understood to be zero outside T i (P i ) for all i = 1, · · · , q). Here and above, χ A is the characteristic function of the set A.
Proof. By the uniqueness of the disintegration (see Lemma 3.2 ) to prove Lemma 4.1, is enough to prove the following equation for a measurable set E ⊂ Σ. To do it, let us define the sets The following properties can be easily proven: Using the change of variables γ = T i (β) and the definition of ν γ (see (22) And the proof is done. Now, if µ ∈ L 1 , appliying the above Lemma to µ + and µ − we directly get 4.2. Proposition. Let γ ∈ F s be a stable leaf. Let us define the map F γ : N 2 −→ N 2 by F γ = π y • F | γ • π −1 γ,y . Then, for each µ ∈ L 1 and for almost all γ ∈ N 1 (interpreted as the quotient space of leaves) it holds (24) (

Basic properties of the norms and convergence to equilibrium
In this section, we show important properties of the norms and their behavior with respect to the transfer operator. In particular, we show that the L 1 norm is weakly contracted by the transfer operator. We prove Lasota-Yorke like inequalities for the strong norms and exponential convergence to equilibrium statements. All these properties will be used in next section to prove a spectral gap statement for the transfer operator. 5.1. Proposition (The weak norm is weakly contracted by F * ). If µ ∈ L 1 then In the proof of the proposition we will use the following lemma about the behavior of the || · || W norm (see equation (9)) after a contraction. Essentially it says that a contraction cannot increase the || · || W norm.

5.2.
Lemma. For every µ ∈ AB and a stable leaf γ ∈ F s , it holds taking the supremum over |g| ∞ ≤ 1 and Lip(g) ≤ 1 we finish the proof of the inequality. Equation (27) is trivial since if µ is a probability measure.
Now we are ready to prove Proposition 5.1.

Proof. (of Proposition 5.1 )
In the following we consider, for all i, the change of variable γ = T i (α). Thus, Lemma 5.2 and equation (24) yield The following proposition shows a regularizing action of the transfer operator with respect to the strong norm. Such inequalities are usually called Lasota-Yorke or Doeblin-Fortet inequalities.
Before the proof of the proposition we prove a preliminary Lemma 5.4. Lemma. Let k, β 0 and C be the constants of assumption T3.3, then there is C > 0 such that for all µ ∈ S 1 , it holds Proof. (of Lemma 5.4 ) Firstly, we recall that φ x is the marginal density of the

Proof. (of Proposition 5.3 )
Note that, iterating one time the inequality (29), we get Thus, for all s ∈ N, we have Therefore, for all s ∈ R, it holds For a given n ∈ N, let n = q n k + r n , where 0 ≤ r n ≤ k. Since PT : 5.1. Convergence to equilibrium. In general, we say that the a transfer operator L has convergence to equilibrium with at least speed Φ and with respect to norms || · || s and || · || w , if for each In this chapter, we prove that F has exponential convergence to equilibrium. This is weaker with respect to spectral gap. However, the spectral gap follows from the above Lasota-Yorke inequality and the convergence to equilibrium. To do it, we need some preliminary lemma and the following is somewhat similar to Lemma 5.2 considering the behaviour of the || · || W norm after a contraction. It gives a finer estimate for zero average measures.
Iterating (32) we get the following corollary.

5.7.
Corollary. For all signed measure µ ∈ L 1 it holds Let us consider the set of zero average measures in S 1 defined by This allows us to apply Theorem 2.1 in the proof of the next proposition.
5.8. Proposition (Exponential convergence to equilibrium). There exist D 2 ∈ R and 0 < β 1 < 1 such that, for every signed measure µ ∈ V s , it holds for all n ≥ 1.

5.9.
Remark. We remark that the rate of convergence to equilibrium, β 1 , for the map F found above, is directly related to the rate of contraction, α, of the stable foliation, and on the rate of convergence to equilibrium, r, of the induced basis map T (see equation 5). More precisely, Similarly, we have an explicit estimate for the constant D 2 , provided we have an estimate for D in the basis map 7 . Now recall we denoted by ψ x the unique T -invariant density in S − (see T3.4). Following the construction exposed in [29] This motivates the following proposition.
5.10. Proposition. The unique invariant measure for the system F : Then µ 0 ∈ L 1 . By construction, ψ x ∈ S − . Then µ 0 ∈ S 1 . And we are done.
If N1 is satisfied, we have | · | ∞ ≤ | · | s . Suppose that g : N 2 −→ R is a Lipschitz function such that |g| ∞ ≤ 1 and L(g) ≤ 1. Then, it holds In this section we consider an L ∞ like anisotropic norm. We show how a Lasota Yorke inequality can be proved for this norm too.
7 It can be difficult to find a sharp estimate for D. An approach allowing to find some useful upper estimates is shown in [15] Proof. Let T i be the branches of T , for all i = 1 · · · q. Applying Lemma 5.5 on the third line below, we have Hence, taking the supremum on γ, we finish the proof of the statement.
Applying the last lemma to F * n instead of F one obtains.

Spectral gap
In this section, we prove a spectral gap statement for the transfer operator applied to our strong spaces. For this, we will directly use the properties proved in the previous section, and this will give a kind of constructive proof. We remark that, we cannot appy the traditional Hennion, or Ionescu-Tulcea and Marinescu's approach to our function spaces because there is no compact immersion of the strong space into the weak one. This comes from the fact that we are considering the same "dual of Lipschitz"distance in the contracting direction for both spaces. 6.1. Theorem (Spectral gap on S 1 ). If F satisfies G1, T1,...,T3.4 given at beginning of section 2, then the operator F * : S 1 −→ S 1 can be written as where a) P is a projection i.e. P 2 = P and dim Im(P) = 1; b) there are 0 < ξ < 1 and K > 0 such that 8 c) P N = N P = 0.
Proof. First, let us show there exist 0 < ξ < 1 and K 1 > 0 such that, for all n ≥ 1, it holds Indeed, consider µ ∈ V s (see equation (33)) s.t. ||µ|| S 1 ≤ 1 and for a given n ∈ N let m and 0 ≤ d ≤ 1 be the coefficients of the division of n by 2, i.e. n = 2m + d. Thus m = n−d 2 . By the Lasota-Yorke inequality (Proposition 5.3) we have the uniform bound || F * n µ|| S 1 ≤ B 2 + A for all n ≥ 1. Moreover, by Propositions 5.8 and 5.1 there is some D 2 such that it holds (below, let λ 0 be defined by λ 0 = max{β 1 , λ}) . Thus, we arrive at Now, recall that F * : S 1 −→ S 1 has an unique fixed point µ 0 . Consider the operator P : ] is the space spanned by µ 0 ), defined by P(µ) = µ(Σ)µ 0 . By definition P is a projection. Define the operator S : S 1 −→ V s , by S(µ) = µ − P(µ) ∀ µ ∈ S 1 . Thus, we set N = F * • S and observe that, by definition, P N = N P = 0 and F * = P + N. Moreover, N n (µ) = F * n (S(µ)) for all n ≥ 1. Since S is bounded and S(µ) ∈ V s we get, by (35), || N n (µ)|| S 1 ≤ ξ n K||µ|| S 1 , for all n ≥ 1, where K = K 1 || S || S 1 →S 1 . 8 We remark that, by this reason, the spectral radius of N satisfies ρ(N) < 1, where N is the extension of N to S 1 (the completion of S 1 ). This gives us spectral gap, in the usual sense, for the operator F : S 1 −→ S 1 . The same remark holds for Theorem ??.
In the same way, using the L ∞ Lasota Yorke inequality of Proposition 5.13, it is possible to obtain spectral gap on the L ∞ like space, we omit the proof which is essentially the same as above: 6.2. Theorem (Spectral gap on S ∞ ). If F satisfies the assumptions G1, T 1, ..., T 3.4 and N 1, then the operator F * : S ∞ −→ S ∞ can be written as where a) P is a projection i.e. P 2 = P and dim Im(P) = 1; b) there are 0 < ξ 1 < 1 and 6.3. Remark. We remark the "gap", ξ, for the map F found in Theorem 6.1, is directly related to the coefficients of the Lasota-Yorke inequality and the rate of convergence to equilibrium of F found before (see Remark 5.9). More precisely, ξ = max{λ, β 1 }. We remark that, from the above proof we also have an explicit estimate for K in the exponential convergence, while many classical approaches are not suitable for this.

Application to Lorenz-like maps
In this section, we apply Theorems 6.1 and 6.2 to a large class of maps which are Poincaré maps for suitable sections of Lorenz-like flows. In these systems (see e.g [4]), it can be proved that there is a two dimensional Poincaré section Σ which can be supposed to be a rectangle, whose return map We remark that, the notion of universal bounded p-variation var p is a generalization of the usual notion of bounded variation. It is a weaker notion, allowing piecewise Holder functions. This notion is adapted to maps having C 1+α regularity.
From these properties, it follows ( [18]) that we can define a suitable strong space (the space S − in T3.1) for the Perron-Frobenius operator P T associated to such a T L , in a way that it satisfies the assumptions T 1, ..., T 3.3 and N 1. In this case, supposing a property like T 3.4 then we can apply our results. For this, let us introduce a suitable space of generalized bounded variation functions with respect to the Lebesgue measure: BV 1, 1 p . The functions of universal generalized bounded variation are included in this weaker space (for more details and results see [18], in particular Lemma 2.7 for a comparison of the two spaces). A piecewise expanding map satisfying assumptions (P'1) and (P'2) has an invariant measure with density in this weaker space, moreover the transfer operator restricted to this space satisfies a Lasota-Yorke inequality and other interesting properties, as we will see in the following.
7.1. Definition. For an arbitrary function h : I −→ C and ǫ > 0 define osc(h, B ǫ (x)) : where B ǫ (x) denotes the open ball of center x and radius ǫ and the essential supremum is taken with respect to the product measure m 2 on I 2 . Also define the real function osc 1 (h, ǫ), on the variable ǫ, by
7.6. Theorem (Spectral gap for BV 1, 1 p ). If FL satisfies assumptions G1, T 1,T 2, T 3.4, P ′ 1 and P ′ 2, then the operator F * L : BV 1, 1 p −→ BV 1, 1 p can be written as F * L = P + N where a) P is a projection i.e. P 2 = P and dim Im(P) = 1; b) there are 0 < ξ < 1 and K > 0 such that for all µ ∈ BV 1, 1 We can get the same kind of results for stronger L ∞ like norms. Let us consider a) P is a projection i.e. P 2 = P and dim Im(P) = 1; b) there are 0 < ξ 1 < 1 and K 2 > 0 such that for all µ ∈ BV ∞ c) P N = N P = 0.

Quantitative Statistical Stability
Through this section, we consider small perturbations of the transfer operator of a given system and try to study the dependence of the physical invariant measure with respect to the perturbation. A classical tool that can be applied for this type of problems is the Keller-Liverani stability theorem [19]. Since in our setting the strong space is not compactly immersed in the weak space, we cannot directly apply it. We will use another approach giving us precise bounds on the statistical stability. In this section, this approach will be applied to a class of Lorenz-like maps with slightly stronger regularity assumptions than used in Section 7.
The following is a general quantitative result relating the stability of the invariant measure for a uniform family of operators and the convergence to equilibrium. Let L be a transfer operator acting on two vector subspaces of signed measures on X, L : (B s , || · || s ) −→ (B s , || · || s ) and L : (B w , || · || w ) −→ (B w , || · || w ) endowed with two norms, the strong norm || · || s on B s , and the weak norm || · || w on B w , such that || · || s ≥ || · || w . Suppose that where SB(X) denotes the space of signed Borel measures on X. UF2 Lδ approximates L0 when δ is small in the following sense: there is C ∈ R + such that: UF3 L0 has exponential convergence to equilibrium with respect to the norms || · || s and || · || w : there exists 0 < ρ 2 < 1 and C 2 > 0 such that for all f ∈ V s it holds || L n 0 f || w ≤ ρ n 2 C 2 ||f || s ; UF4 The iterates of the operators are uniformly bounded for the weak norm: there exists M 2 > 0 such that ∀δ, n, g ∈ B s it holds || L n δ g|| w ≤ M 2 ||g|| w .
We will see, under these assumptions we can ensure that the invariant measure of the system varies continuously (in the weak norm) when L0 is perturbed to Lδ, for small values of δ. Moreover, the modulus of continuity can be estimated.
Let us state a general lemma on the stability of fixed points satisfying certain assumptions. Let us consider two operators L0 and Lδ preserving a normed space of signed measures B ⊆SB(X) with norm || · || B . Suppose that f 0 , f δ ∈ B are fixed points, respectively of L0 and Lδ.
Proof. The proof is a direct computation C i by item a), and then Now, let us apply the statement to our family of operators satisfying assumptions UF 1,...,4, supposing B w = B. We have the following Proof. Let us apply Lemma 8.2. By UF2, Hence, By the exponential convergence to equilibrium of L0 (UF3), there exists 0 < ρ 2 < 1 and C 2 > 0 such that (recalling that by UF1 8.1. Quantitative stability of Lorenz-like maps. Here we apply the general results on uniform families of operators to a suitable family of bounded variation Lorenz-like maps. We consider maps as defined in Section 7, with some further assumptions. y)), is said to be a BV Lorenz-like map if it satisfies (1) There are H ≥ 0 and a partition P ′ = {J i := (b i−1 , b i ), i = 1, · · · , d} of I such that for all x 1 , x 2 ∈ J i and for all y ∈ I the following inequality holds (2) F L satisfy property G1 (hence is uniformly contracting on each leaf γ with rate of contraction α); (3) T L : I → I is a piecewise expanding map satisfying the assumptions given in the following definition 8.5.
The following definition characterizes a class of piecewise expanding maps of the interval with bounded variation derivative T L : I −→ I which is a subclass of the ones considered in section 7.0.1. 3) T L satisfies T3.4; In particular we assume that T Li and g i admit a continuous extension to P i = [a i−1 , a i ] for all i = 1, · · · , q. 8.6. Remark. The definition 8.5 allows infinite derivative for T L at the extreme points of its regularity intervals.
We have seen that a positive measure on the square, [0, 1] 2 , can be disintegrated along the stable leaves F s in a way that we can see it as a family of positive measures on the interval, {µ| γ } γ∈F s . Since there is a one-to-one correspondence between F s and [0, 1], this defines a path in the space of positive measures, [0, 1] −→ SB(I).
In the following, when no ambiguity is possible we will consider informally Γ µ itself as a path. Let us call the set on which Γ ω µ is defined by I Γ ω µ . 8.10. Definition. Let P = P(Γ ω µ ) be a finite sequence P = {x i } n i=1 ⊂ I Γ ω µ and define the variation of Γ ω µ with respect to P as (denote γ i := γ xi ) where we recall || · || W is the Wasserstein-like norm defined by equation (9). Finally we define the variation of Γ ω µ by taking the supremum over the sequences, as Var(Γ ω µ ) := sup P Var(Γ ω µ , P).

8.11.
Remark. For an interval η ⊂ I, we define , where η is the closure of η.
Now we are ready to state a lemma estimating the regularity of the iterates F * n (m). Next result is a Lasota-Yorke-like inequality where the strong semi-norm is the variation Var(µ) defined in 8.13. This is our main tool to estimate the regularity of the invariant measure of a BV Lorenz-like map. The proof will be postponed to the appendix (see Proposition 9.13). , y)) be a BV Lorenz-like map. Then, there are C 0 , 0 < λ 0 < 1 and k ∈ N such that for all µ ∈ BV + and all n ≥ 1 it holds (denote F := F k ) Var( F * n µ) ≤ C 0 λ n 0 Var(µ) + C 0 |φ x | 1,1 . A precise estimate for C 0 can be found in equation (69). Remember, by Proposition 5.10, a Lorenz-like map has an invariant measure µ 0 ∈ S ∞ . 8.16. Proposition. Let F L (x, y) = (T L (x), G L (x, y)) be BV Lorenz-like map and suppose that F L has an unique invariant probability measure µ 0 ∈ BV ∞ 1,1 . Then µ 0 ∈ BV + and Var(µ 0 ) ≤ 2C 0 .
Proof. Let F := F k L where k comes from Proposition 8.15. Then µ 0 ∈ BV ∞ 1,1 is the unique F -invariant probability measure in BV ∞ 1,1 . Consider the Lebesgue measure m and the iterates F * n (m). By Theorem 7.7, these iterates converge to µ 0 in L ∞ .
8.17. Remark. We remark that, Proposition 8.16 is an estimation of the regularity of the disintegration of µ 0 . Similar results are presented in [16] and [11].
The proof of the following proposition is postponed to the appendix (see Proposition 9.14). (8.7)) and let f δ be the unique F δ -invariant probability in BV ∞ 1,1 . Then, there exists C 0 > 0 such that for all δ ∈ [0, 1).
We are now ready to prove the following 8.19. Proposition (to obtain UF2).
Since F * is a contraction for the weak norm, we have Now, let us estimate the first summand of (54) by estimating the integral where µ = 1 A f δ . Denote by T 0,i , with 0 ≤ i ≤ q, the branches of T 0 defined in the sets P i ∈ P and set T δ,i = T δ | Pi∩A . These functions will play the role of the branches for T δ . Since in A, T 0 = T δ • σ δ (where σ δ is the diffeomorphism in the definition of the Skorokhod distance), then T δ,i are invertible. Then Let us now consider T 0 (P i ∩A), T δ (P i ∩A) and remark that T 0 (P i ∩A) = σ δ (T δ (P i ∩ A)) where σ δ is a diffeomorphism near to the identity. Let us denote And since there is K 1 such that m(C i ) ≤ K 1 δ, we get In order to estimate O 1 , we note that The two summands will be treated separately. Let us denote µ| γ = π * γ,y µ γ (note that µ| γ = φ µ (γ)µ| γ and µ| γ is a probability measure).
To estimate I b (γ), we have By [28], Lemma 11.2.1, we get Now, let us estimate the integral of the second summand Let us make the change of variable γ = T δ,i (β).
Once this is done, we have all the ingredients to apply Proposition 8.3 and obtain the quantitative estimation.  In this section, we obtain Proposition 8.15 as a particular case of Theorem 9.2. Note that, for all µ ∈ BV + it holds ||µ|| 1 = |φ x | 1 and ||µ|| ∞ = |φ x | ∞ , where . We also remark, for each µ ∈ BV + we have φ x ∈ BV 1,1 .
Since preliminaries results are necessary, we postponed the proof of the next theorem to the end of the section. Then, there are C 0 , 0 < λ 0 < 1 and k ∈ N such that for all µ ∈ BV + and all n ≥ 1 it holds (denote F := F k ) T i ′ . For all n ≥ 1, let P (n) be the partition of I s.t. P (n) (x) = P (n) (y) if and only if P (1) (T j (x)) = P (1) (T j (y)) for all j = 0, · · · , n, where P (1) = P (see definition 8.5). Given P ∈ P (n) , define g (n) P = 1 |T n ′ |P | . Item 2) implies that there exists C 1 > 0 and 0 < θ < 1 s.t.
9.9. Lemma. For all Γ µ , where µ ∈ BV + , and all P ∈ P it holds Proof. Consider (γ i ) n i=1 ⊂ P such that γ 1 ≤ · · · ≤ γ n . By Lemma 9.6, for every i there is y i such that Then, We finish the proof taking the supremum over (γ i ) n i .
We a ready to prove Theroem 9.2.