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Escape Rates Formulae and Metastability for Randomly perturbed maps

We provide escape rates formulae for piecewise expanding interval maps with `random holes'. Then we obtain rigorous approximations of invariant densities of randomly perturbed metabstable interval maps. We show that our escape rates formulae can be used to approximate limits of invariant densities of randomly perturbed metastable systems.


Introduction
A dynamical system is called open if there is a subset in the phase space, called a hole, such that whenever an orbit lands in it, the dynamics of this obit is terminated. In open dynamical systems, long-term statistics are described by a conditionally invariant measure and its related escape rate, measuring the mass lost from the system per unit time [13,15]. For the past three years there has been a considerable interest in describing the escape rate of an open system as a function of the hole's position and size. In [8] it was observed that for the doubling map, given two holes, H 1 and H 2 of the same fixed size, each centred around a periodic point, say x 1 , x 2 respectively, with the period of x 1 smaller than that of x 2 , then the escape rate through H 1 is bigger than that through H 2 . Later in [27] it was shown that, by shrinking the hole to a point, say x 0 , the escape rate depends on two things: i) whether x 0 is periodic or not; ii) the invariant density of the corresponding closed system. Following the success of [8,27], other researchers studied this phenomenon for different types of systems [1,12,17] 1 . These results have lead to insights in studying metastable dynamical systems which behave approximately like a collection of open systems: the infrequent transitions between almost invariant regions in a metastable system are similar to infrequent escapes from associated open systems [22,16,19,27].
This research topic is currently very active in ergodic theory and dynamical systems. In part, this is due to the interesting applications of open and metastable dynamical systems in physical sciences. For instance, open dynamical systems are used to study transport in heat conduction [21]. They also play an important in astronomy [33]. In molecular dynamics, almost invariant regions of metastable systems are used to identify sets where stable molecular conformations occur [31]. In astrodynamics, metastable systems are used to recognize regions from which asteroids escape is infrequent [11]. Moreover, metastable dynamical systems have been recently used to develop realistic models of atmospheric and ocean circulation [10,18,30]. In these models, it is believed that metastable states 2 lie behind long term global circulation patterns, and form large scale barriers to transport.
In this paper we first study piecewise smooth and expanding interval maps with 'random holes'. We prove that the random open system admits an absolutely continuous conditionally stationary measure (accsm). Moreover we obtain escape rates formulae (first order approximations) depending on the position of the holes. An older escape rate formula for randomly perturbed maps with specific holes positions and a specific distribution on the noise space was obtained in [9]. Our result is in the spirt of the recent trend of describing the escape rate as a function of the position of the hole [1,8,17,27]. In particular, our result generalizes Keller-Liverani escape rates formulae [27] which were obtained for deterministic perturbations.
In the second part of the paper we study random perturbations of interval maps that initially admit exactly two invariant ergodic densities. However, under random perturbations that generate random holes and allow leakage of mass between the two initially ergodic subsystems, the random system admits a unique stationary density. We show that such a density for the metastable random system can be approximated in the L 1 norm (with respect to the ambient Lebesgue measure m), by a particular convex combination of the two invariant ergodic densities of the initial system. In particular, we show that the ratio of the weights in the convex combination is equal to the ratio of the averages of the measures of the left and right random holes. Moreover, as a by product of our escape rates formulae, we show that these weights can be also identified as the ratio of the escape rates from the left and right random open systems.
We finish the introduction by addressing possible generalizations of our work. We foresee two natural generalizations. The first generalization concerns the extension to piecewise expanding maps in higher dimensions. Two difficulties arise in this setting. First, obtaining a Lasota-Yorke inequality for the operator associated with the open system with random holes, see Lemma 2.6 would require a more careful control of the variation, whatever the Banach space B that operator acts on is, of the characteristic function along the boundaries of the partitions, which are now codimension-1 piecewise smooth submanifolds and not points as in the setting of this paper. The second difficulty is to get local smoothness of elements of B around the holes similar to Lemma 1 of [22]. We believe that the latter property is fundamental to obtain results on approximating invariant densities of metastable systems, even in the one-dimensional case.
The second generalization is the extension to non-uniformly expanding maps. Metastability for deterministic maps with indifferent fixed points was treated by 2 Metastable states correspond to eigenvalues of a transfer operator which are very close to 1, a phenomenon which is very transparent in the metastable systems of this paper. us in [3]. One of the main tools we used there was to reduce the study of metastability to an induced subsystem where the map is uniformly expanding, and then pullback the result to the original system. It is not clear how this technique can be implemented in the presence of noise. A few recent results [2,32] used induction to achieve stochastic stability in some classes of one-dimensional maps. It would be interesting to explore the possibility of apply those techniques to metastability. Supposing the analogous of our Theorem 3.4 can be proved in the random case, it would be also interesting to explore if something similar to Corollary 3.5 (see section 3.3) would still hold. In this paper, the latter is related to the exponential escape rate which is in turn associated to the existence of a spectral gap of the transfer operator. This is a 'luxury' that is not available in a non-uniformly hyperbolic setting.
In section 2 we obtain escape rates formulae for expanding interval maps with random holes. Our main result in this section is Theorem 2.1. In section 3.3 we study rigorous approximations in the L 1 -norm of invariant densities of randomly perturbed metastable maps. Our main result in this section is Theorem 3.4.

Escape Rate Formulae For Randomly Perturbed Maps
2.1. Notation. Throughout the paper we use the following notation: We denote by BV (I) the space of functions of bounded variation on I equipped with the norm || · || BV = V (·) + || · || 1 , where || · || 1 is the L 1 norm with respect to m.

2.2.
Piecewise expanding maps. Let T : I → I be a map which satisfies the following conditions: (O1) There exists a partition of I, which consists of intervals T preserves a unique acim 3 µ which is equivalent to m. Moreover, the system (I, µ, T ) is mixing.
2.3. Random Holes. Let (ω k ) k∈N be an i.i.d. stochastic process with values in the interval Ω ε = [0, ε], ε > 0, and with probability distribution θ ε . We fix z ∈ (0, 1) and we associate with each ω ∈ Ω ε an interval H ω such that z ∈ H ω ⊆ H ε , and we assume that H ε ⊆ H ε for ε ≤ ε . Further, (O4) We assume that T is continuous at z; this assumption will be explicitly used in the proof of Theorem 2.1. In order to apply the results of this section to section 3.3, we will also assume that the density ρ is continuous at z. Figure 1. A map with random holes H ω1 and H ω2 sitting inside H ε . Note that H ω1 and H ω2 are not nested.

2.4.
Transfer operator of the random open system. Our goal in this section is to study the existence of absolutely continuous conditionally stationary measures (accsm) and their associated escape rates through the random holes H ω around a given point z under the dynamics of T . An example of an interval map with random holes is shown in Figure 1. We set and define for f ∈ L 1 (I, B, m) where L 0 is the transfer operator (Perron-Frobenius) [4,7] associated with T ; i.e., for f ∈ L 1 (I, B, m) we have The transfer operatorL ε will be used to prove that the random open system admits an accsm with exponential escape rate. By using the fact that for any measurable set A and integrable function f where θ ∞ ε := Π ∞ i=1 θ ε andω := (ω 1 , ω 2 , . . . ).
2.5. Statement of the main result of section 2. The following theorem provides random versions of Keller-Liverani escape rates formulae [27].
Theorem 2.1. For sufficiently small ε > 0, there exists an e ε , 0 < e ε < 1, and a g ∈ BV (I), g ε (x) > 0, with I g ε, dm = 1, such that Moreover, (2) If z is a periodic point of T of minimal period p, T p is C 1 in a neighbourhood of z, and the probability distribution θ ε on the noise space Ω ε satisfies the following condition: ∃ υ > 1 such that Remark 2.2.
(1) In (2) of Theorem 2.1, condition (C) means that for a fixed ε > 0 most of the random holes H ω are not too small when compared to the size of H ε . Notice that the two "υ" in the condition can be different, provided they remain larger than 1. We considered the same υ for simplicity 4 . Condition (C) is needed to insure a lower bound on the escape rate in case (2) of Theorem 2.1. (2) For the existence of the accsm we required ε to be sufficiently small since we use the perturbation result of [26] in the proof our Lemma 2.7. The strict positivity of the density g ε also follows from [26] since µ is equivalent to m.
where A ⊆ I is a measurable set. By using the fact thatL ε g ε = e ε g ε and the definition ofL ε , we obtain that ν ε satisfies the following: Definition 2.4. We will call − ln e the escape rate for the random system, and ν the absolutely continuous conditionally stationary measure.
Remark 2.5. It is interesting to remark that the items (1) and (2) of Theorem 2.1 provide explicit perturbation formulae for the leading eigenvalue of the operator L ε . This operator enjoys all the properties of what Keller [25] recently called rare events Perron-Frobenius operators.
2.6. Proof of Theorem 2.1. We start this subsection by proving a uniform Lasota-Yorke inequality forL ε and L 0 . A deterministic version of Lemma 2.6 can be found in [29].
Proof. We begin to compute the total variation, leaving the L 1 estimate at the end.
In particular, Now, let us consider a fixed random path of length n, (ω 1 , ω 2 , . . . , ω n ), f ∈ BV (I) and define g := f 1 Xω 1 ∩T −1 Xω 2 ∩···∩T −(n−1) Xω n and observe that g ∈ BV (I). Then using (2.6) and noticing that consists of at most n + 1 connected components, we obtain (2.8) Summing over A ∈ Z n in (2.8) we obtain Since the inequality in (2.9) does not depend on ω and we also have The estimate on ||L n ε f || 1 is easy. Indeed, this can be done by splitting f into its positive and negative parts and by using the linearity of the transfer operator. Therefore, we may suppose that f is non negative. This allows us to interchange the integrals w.r.t. the Lebesgue measure and θ ∞ ε and to use duality. In conclusion we get Since there exists n 0 and γ ∈ (κ, 1) such that 2κ n0 (2n 0 + 3) ≤ γ n0 , we can choose A := 2n 0 + 3, B := sup n≤n0 1 min A∈Z n m(A) 2(n + 1) 2 1−γ n 0 and use (2.10), (2.7) to obtain a uniform the Lasota-Yorke inequality forL ε and L 0 .
(2) If for each integer k ≥ 0 the following limit: Proof. By condition (O3), for sufficiently small ε > 0, ∆ ε > 0. Thus, Part (2) of the lemma follows from the abstract perturbation result of [27], since we will show in the proof of Theorem 2.1 that for each k ≥ 0 the limit in (2) exists.
Proof. (Proof of Theorem 2.1) We start by computing q k,ε and show that the limit in (2) of Lemma 2.8 holds. Throughout the computation in (2.16) below, we will use repeatedly the fact that for any measurable set A and integrable function f we Consequently, by (2.11) and (2.2), we obtain Now, if z is not a periodic point of T , since T is continuous at z, for sufficiently small ε > 0, Therefore, (2.16) implies q k,ε = q k = 0. This proves (1) of the theorem.
To prove (2) of the theorem, since z is a periodic point of minimal period p and the density ρ is essentially bounded from below, the quantity we have to compute reduces to Since T p is continuous around z we can suppose that, by taking small enough, all the H ω , |ω| ≤ , will be contained in a neighborhood of z where T p is continuous and monotone. This in particular implies the following fact that we will use later on: if we call H ω,p the unique connected component of T −p H ω which contains z, it will be the only connected component of T −p H ω which intersects any other H ω .
By the differentiability assumption on T p and the local change of variable we have By (2.19) and the continuity of DT p at z we obtain For the lower bound, since T p is expanding, we have Therefore, Let us consider the second expression on the right hand side of (2.22) and show that it vanishes when ε goes to zero. First, we have which goes to zero when ε tends to 0. Consequently, taking the limit as ε goes to 0 in (2.22), by the continuity of DT p at z, we obtain Thus (2) of the theorem follows by (2.20) and (2.23).

Metastability of randomly perturbed maps
In this section we study random perturbations of Lasota-Yorke maps. We assume that the initial system admits exactly two ergodic invariant densities. Then under random perturbations which allow leakage of mass through random holes, we will show that the system admits a unique stationary density. Our goal is to show that such a density can be approximated in the L 1 -norm by a particular convex combination of the two invariant densities of the initial system. A deterministic Lasota-Yorke system of this type was studied in [22]. Deterministic intermittent systems of this type were studied in [3]. Other recent results on metastable random dynamical systems have been obtained in [20,24]. We will establish a close link between the escape rate formulae which we obtained in the previous section and invariant densities of randomly perturbed metastable systems. We first introduce the class of maps of this section.
3.1. The initial system. Let T : I → I be a map which satisfies the following conditions: (A1) There exists a partition of I, which consists of intervals (A5) T admits exactly two ergodic a.c.i.ms µ * , such that each µ * is supported on I * and the corresponding density ρ * is positive at each of the points of H 0 ∩ I * Remark 3.1. As shown in [22], assumption (A4) guarantees that the invariant densities of the two ergodic measures are continuous at each of the infinitesimal holes. Assumption (A2), more precisely the fact that β −1 0 is strictly bigger than 2 instead of 1, is sufficient to get the uniform Lasota-Yorke inequality of Proposition 3.2, as is explained in section 4.2 of [22]. Finally, we consider T to be bi-valued at the points c i,0 of discontinuity in C 0 and we take T (c i,0 ) be both values obtained as x tends to c i,0 from either side, and T (c i,0 ±) will be the corresponding right and left limits.
An example of a map T 0 satisfying the above assumption is shown in Figure 2. We denote the transfer operator (Perron-Frobenius) [4,7] associated with T by L 0 . Figure 2. A typical example of the initial system T .

3.2.
Random perturbations on a finite noise space. As in the previous section, let (ω k ) k∈N be an i.i.d. stochastic process with values in Ω ε and with probability distribution θ ε . However, we assume in this section that Ω ε = {s 0 , s 1 , . . . , s L } is a finite set, and θ ε is an atomic measure on Ω ε 5 . We associate with each ω ∈ Ω ε a map T ω : I → I with T 0 = T s0 , T ε = T s L , and we consider the random orbit starting from the point x and generated by the realization ω n = (ω 1 , ω 2 , · · · , ω n ), 5 In fact, the assumption that the set Ωε is a finite set is not need in some of the proofs of this section. In particular in Lemmas 3.7 and 3.8. However, this assumption is needed to be able to identify the jumps in the stationary density ρε of the random system. It seems that the assumption of this section that Ωε is a finite set is the best one can do when dealing with approximating the invariant density of the random metastable system. See Remark 3.10 where we explain why the problem becomes intractable if one relaxes this assumption even to the case where Ωε is countable but not finite. defined as : T ω n := T ωn • · · · • T ω1 (x) (random transformations). This defines a Markov process T ε with transition function where A ∈ B(I), x ∈ I and 1 A is the indicator function of a set A. The transition function induces an operator U * ε which acts on measures Λ on (I, B(I)) as: where U ε is the random evolution operator acting on L ∞ m functions g: A measure µ ε on (I, B(I)) is called a T ε -stationary measure if and only if, for any A ∈ B(I), . We are interested in studying the metastability of T ε -stationary measures which are absolutely continuous with respect to m: let us call them acsm. By (3.1), one can define the transfer operator L ε (Perron-Frobenius) acting on L 1 (I, B(I), m) by: which satisfies the duality condition where g is in some subset of L ∞ m and L ω is the transfer operator associated with T ω . In the present setting g ∈ BV (I); i.e., a function of bounded variation. It is well know that µ ε := ρ ε m is a T ε -acsm if and only if L ε ρ ε = ρ ε ; i.e., ρ ε is a T ε -invariant density. For each ω ∈ Ω ε , we assume that T ω satisfies the following conditions: (B1) There exists a partition of I, which consists of intervals {I i,ω } q i=1 , I i,ω ∩I j,ω = ∅ for i = j,Ī i,ω := [c i,ω , c i+1,ω ] such that (i) for each i and for all 0 ≤ ω ≤ ε, ε small enough, we have that (the quantity δ was introduced in the assumption (A1) above): ; whenever c i,0 = 0 and c i+1 , 0 = 1. In this way we have established a one-to-one correspondence between the unperturbed and the perturbed boundary points of I i and I i,ω . If c i,0 and c i,ω are two such (left or right) corresponding points we will ask that ∀i and ∀ω ∈ Ω ε : (ii) The map T ω is locally injective over the closed intervals I i,ω , of class C 2 in their interiors, and expanding with inf ω,x |T ω x| ≥ β > 2. Moreover, ∀ω ∈ Ω, ∀i = 1, · · · , q and ∀x ∈ [c i,0 − δ, c i+1,0 + δ] we have The last condition, as explained in section 2.4 of [22] does not allow the appearance of other holes, besides those around the infinitesimal holes, in the neighborhood of b. These conditions are satisfied when the noise is, for instance, additive: T ω (x) = T (x) + ω. It is always well defined on the circle by taking the mod-1 unfolding; however, on the interval we should consider maps T for which the images |T ω (x)| lie in the unit interval for all ω ∈ Ω ε . In both cases the intervals of local monotonicity will be always the same and the differences between the perturbed and unperturbed images are uniformly bounded by ε. Another example of a T ω is shown in Figure 3. Under the above assumptions it is well known that (see for instance [6]) Proposition 3.2. For sufficiently small ε > 0, there exists a η ∈ (0, 1) and a B ∈ (0, ∞), such that for any f ∈ BV (I) and n ≥ 1, we have L n • f BV ≤ η n f BV + B||f || 1 , where • ∈ {0, ε}; i.e., both L 0 and L ε satisfy a uniform Lasota-Yorke inequality as operators on BV (I).
Among other things, Proposition 3.2 implies that 1 is an eigenvalue of L ε and T ε admits a stationary density ρ ε . We further assume assume that: (R1) ρ ε is the unique invariant density of T ε (ergodicity). Remark 3.3. Since in this section we assumed Ω ε to be finite, then condition (R1) is satisfied if one of the maps T ω has a unique acim. See Corollary 2 of [23]. For instance, condition (R1) is satisfied whenever the deterministic map T ε is topologically mixing.
3.3. Random holes in the perturbed system. We are interested in perturbations of T which produce "leakage" of mass from I l to I r and vice versa. For this purpose, for each ω ∈ Ω ε we define the following sets: H r,ω := I r ∩ T −1 ω (I l ). The sets H l,ω and H r,ω are called the "left hole" and the "right hole", respectively, of the map T ω . Thus, when T ω allows leakage of mass from I l to I r , this leakage occurs when orbits of T ω fall in the set H l,ω . Similarly, when T ω allows leakage of mass from I r to I l , this leakage occurs when orbits of T ω fall in the set H r,ω .
For each of the left and right 'random open systems', we define a transfer operator which will be used to find the exponential escape rates from I l and I r . For this purpose we set X l,ω := I l \ H l,ω , and define for f ∈ L 1 (I l , B(I l ), m b ) where L l,0 is the transfer operator associated with the map T | I l := T l : I l → I l .
In a similar way, we can define a transfer operatorL r,ε associated with the right 'random open system'.

3.4.
Statement of the main result of section 3.3. Using Proposition 3.2 and compactness arguments, it is well known that ρ ε converges in the L 1 -norm to a convex combination of ρ l and ρ r . Our main goal in this part of the paper is to explicitly identify the weights in the convex combination. Since the acsm is ergodic, at least one of the two holes will have positive Lebsegue measure; we will suppose without loss of generality that the left hole will have this property. It will also carry positive mass for the left unperturbed invariant measure µ l since we assumed that the densities of µ l and µ r are positive in the neighborhood of the respective holes. We therefore define the limiting averaged holes ratio (l.a.h.r.) by (3.9) l.a.h.r. := lim ε→0 Ωε µ r (H r,ω )dθ ε (ω) Ωε µ l (H l,ω )dθ ε (ω) whenever the limit exists. We obtain the following theorem:  (1) For sufficiently small ε > 0, there exists an 0 < e * ,ε < 1, and a g * ,ε (x) > 0, with I * g * ,ε dm = 1, such that L * ,ε g * ,ε = e * ,ε g * ,ε .
(2) Moreover, Remark 3.6. As it was observed in [22], and this remains true in our case as well, the limit (3.9) surely exists whenever the perturbations open up holes whose size is first order in ε.
We also note that condition (B4) is satisfied, for instance, when the map T is continuous in the neighborhood of the infinitesimal holes and we perturb it with an additive noise.

3.5.
Technical lemmas and the proof of Theorem 3.4.
Let us call I l,ωj the l-domain of injectivity of the map T ωj and call T −1 l,ωj the inverse of T ωj restricted to I l,ωj . By T l,0 we denote the restriction of the unperturbed map to its l-th interval of monotonicity; finally for simplicity of notation we drop the suffix l, r from the intervals of the partition. We have: The sets Ω k1,··· ,kn ω1,··· ,ωn := T −1 k1,ω1 • · · · • T −1 kn−1,ωn−1 I kn,ωn ∩ T −1 k1,ω1 • · · · ∩ T −1 k1,ω1 I k2,ω2 ∩ I k1,ω1 are intervals and they give a partition mod-0 of I = [0, 1]; moreover the image H k1,··· ,kn ω1,··· ,ωn := T ωn • · · · • T ω1 Ω k1,··· ,kn ω1,··· ,ωn for a given n-tuple {k n , · · · , k 1 } is a connected interval. We will prove, by contradiction, that the function Ψ ωn,··· ,ω1 (x) is smooth in a neighborhood of h * , actually C 1 , when h * is in the interior of one or several of the images H k1,··· ,kn ω1,··· ,ωn described above. If Ψ ωn,··· ,ω1 (x) is not smooth in a neighborhood of h * on any interval O ⊃ h * , we can find a sequence (ω n , · · · ω n−l ), 1 ≤ l ≤ n − 1 and at least an endpoint of one of the domain of injectivity of T ω n−l , call it c * ω n−l , such that T ωn • · · · • T ω n−l (c * ω n−l ) ∈ O. We now proceed by induction and we begin by showing that Ψ ω is smooth in a neighborhood of h * for any choice of ω. If not there will a point c l,ω (see above, where we now mean that this point is one of the two boundaries of the interval I l,ω ), such that T ω l c l,ω ∈ (h * − σ /2, h * + σ /2). Let us now take the corresponding point c l,0 of T which will belongs to C 0 . We have |T l,0 (c l,0 ) − T l,ω (c l,ω )| ≤ |T l,0 (c l,0 ) −T l,0 (c l,ω )| + |T l,0 (c l,ω ) − T l,ω (c l,ω )| By the uniform conditions (3.5) and (3.6) we can take ε smaller that a certain ε σ depending on σ such that |T l,0 (c l,0 ) −T l,0 (c l,ω )| ≤ sup l, x∈I l,0 |DT l,0 ||c l,0 − c l,ω | ≤ σ /2 and |T l,0 (c l,ω ) − T l,ω (c l,ω )| ≤ σ /2 and this implies |T l,0 (c l,0 ) − h * | ≤ σ which contradicts the above statement T n C 0 ∩ H 0 = ∅. Fix σ = σ /2; we now continue the induction process by supposing that for any sequence (ω 1 , · · · , ω n ) the function Ψ ωn,··· ,ω1 is smooth on the interval (h * − σ, h * + σ) and we want to prove that the function Ψ ωn+1,··· ,ω1 is still smooth for the sequence (ω 1 , · · · , ω n+1 ). For that it will be enough to repeat the previous argument by noticing that This shows that Ψ ωn,··· ,ω1 (x) is smooth in a neighborhood of h * . We now take two points x, y in the interval (h * − σ, h * + σ); we introduce the notation T −1 ωn := T −1 k1,ω1 • · · · • T −1 kn,ωn and we compute |Ψ ωn,··· ,ω1 (x) − Ψ ωn,··· ,ω1 (y)| ≤ where we skip the characteristic function since both x, y will be in the same Ω k1,··· ,kn ω1,··· ,ωn . We have which follows from our assumptions on the map T and its perturbations, and the sum in (3.13) will be bounded by C 1 times the sum of a geometric series of common ratio β −1 : we call C 2 the upper bound thus found. Therefore, by the mean value theorem, we have where ζ ∈ (x, y). We can replace ξ with x (or y) by a standard distortion argument which works in our case by the assumptions we have on the maps T ω . By modifying the constant C 2 into a constant C 3 which takes into account the distortion factor we finally have (3.14) We now integrate (3.14) over the noise and get, for any x, y ∈ Iσ, is obtained from the Lasota-Yorke inequality in Proposition 3.2. The Lemma finally follows by choosing K = C 3 (1 + B).
In [22], the authors dealt with the deterministic version of this section of our paper. In [22], the notion of the postcritical set of a map was used extensively to identify the location of discontinuities of the invariant density. However, in the random setting, the notion of a postcritical set of T ε does not exist. Following the ideas of our Lemma 3.8 above, we develop an approach which is suitable for our random setting. We first recall a useful representation of a 1-dimensional function of bounded variation 6 . For f ∈ BV (I), choose a version of f with regular discontinuities: for each x, f (x) = (lim y→x − f (y) + lim y→x + f (y))/2. Then, f can be uniquely decomposed as f = f reg + f sal , where the regular term f reg is continuous and of bounded variation, with V (f reg ) ≤ V (f ), while the singular (or saltus) part f sal is the sum of jumps where S is a countable set, and This representation imposes the boundary condition f sal (1) = 0. Moreover, the variation of the singular part satisfies where L k ω = L ω1 · · · L ω k . Note that, by (R1), we have lim n→∞ ||F n,ε − ρ ε || 1 = 0.
(1)S n is a finite set and S is a countable set. Proof.
Since Ω ε is a finite set, (1) follows by definition ofS n and S. (2) is a consequence of (1). (3) follows by definition ofS n .
Remark 3.10. As we have mentioned in footnote 4, if one relaxes the assumption that Ω ε is finite, even to the case where Ω ε is countable but not finite, then intractable complications arise. Indeed, if Ω ε is countable but not finite, one can construct an example of a random metastable system such that in an interval, [u, v], say, each rational point is in the postcritical set of a different T ω . For a system like this, the complement set [0, 1] \S n becomes uncountable, and its intersection with [u, v] will not contain any interval 7 . This makes it impossible to obtain regularity properties of ρ reg ε , the regular part of the stationary density ρ ε .
Lemma 3.11. The discontinuities of F n,ε belong toS n .
Proof. We prove the lemma by contradiction. Namely, let a be a discontinuity point for F n,ε , and suppose that a / ∈S n ,. Then ∀ω and 0 ≤ k ≤ n − 1, we have that Since, for sufficiently small , each L ω , ω ∈ Ω ε , satisfies a Lasota-Yorke inequality with uniform constants (see Proposition 3.2) we concatenate the transfer operators and get the existence of a positive constant C , independent of the realization, for which ||L k ω 1|| ∞ ≤ ||L k ω 1|| BV ≤ C . Therefore we can use (3.15) and apply the Lebesgue dominated convergence theorem to get Thus, a contradiction.
(1) The discontinuity set of ρ ε is subset of S. If we write ρ sal ε := u∈S s u H u , then u∈S,#u>m |s u | ≤ η −m C .
(2) ρ reg ε is Lipschitz continuous. Proof. Since F n,ε is a function of bounded variation, F n,ε can be decomposed as F n,ε = F reg n,ε + F sal n,ε . By Lemma 3.11, the discontinuity set of F n,ε is contained iñ S n ⊂ S. Then we can write the saltus part of F n,ε as F sal n,ε = u∈S s u,n H u .
We first show that #u>m |s u,n | decays exponentially fast in m. To show this, observe that if m ≥ n then u∈S; #u>m |s u,n | = 0 and u∈S; #u>0 Moreover, Now for #u > 1, i.e. u / ∈Ŝ 1 , which means that ∀ω, each preimage T −1 k,ω is continuous at u. Therefore the contributions to the jump s u,n come from the jumps of F n−1,ε and are located at the points v such that T ω v = u, precisely By using a diagonalization argument, we may find a subsequence n j such that for each u ∈ S, s u,nj converges as n j → ∞ to some number, which we call s u . Let us construct the function It is clear that F sal n,ε converges in L 1 to F sal ε and that for each m: u∈S,#u>m |s u | ≤ η −m C . We now show that F sal ε coincides with ρ sal ε and F reg , a limit point of F reg n,ε , coincides with ρ reg ε . Indeed, by (3) of Lemma 3.9, F n,ε is C 1 on the open intervals of [0, 1] \S n . Therefore, on [0, 1] \S n , the derivative of F n,ε coincides with the derivative of F reg n,ε (since F sal n,ε is constant on each interval of [0, 1] \S n ). Then, using the arguments in Lemma 3.8, in particular estimates (3.13) and (3.14), we obtain a uniform estimate on the derivative of F reg n,ε on [0, 1] \S n . By (2) Lemma 3.9, [0, 1] \S n consists of a finite number of intervals. Thus, since by definition F reg n,ε is a continuous function, we conclude that F reg n,ε is uniformly Lipschitz on [0, 1]. Consequently, by the Arzelà-Ascoli theorem, we can find a continuous function F reg such that some subsequence of {F reg nj ,ε } converges in L ∞ , and hence in L 1 , to F reg . By the uniqueness of the decomposition ρ ε = ρ sal ε + ρ reg ε , we get that F reg = ρ reg ε and F sal = ρ sal ε .
We now observe that by the Lasota-Yorke inequality of Proposition 3.2 and the compactness of the unit closed ball of || · || BV in L 1 , there are subsequences of valuesε ∈ Ω ε for which ρε converges in the L 1 norm to the density of an acim on I. Obviously, such a density will surely be a convex combination of ρ l and ρ r . We would like to show that those accumulation points are always the same so that ρ ε will admit a limit when ε → 0. The first step is to characterize the coefficients of the convex combination in terms of the behavior of the density in the neighborhoods of the holes. We first have: Lemma 3.14. Let ρ 0 be the accumulation point of the convergent subsequence ρε withε ∈ Ω ε ; then there exists 0 ≤ α ≤ 1 such that: 8 (1) ρ 0 = αρ l + (1 − α)ρ r ; (2) lim Proof. We argued above that (1) is true. The proof of (3) will be identical to that of (2). Thus, we only prove (2). Observe that for all ω ∈ Ωε we have H l,ω ⊆ H l,ε . Therefore, sup We will prove (3.16) by contradiction. For this purpose we will suppose that the there exists a C > 0 and a subsequence ε → 0 ofε such that for each ε there is an x ε ∈ H l,ε , x ε → h l with |ρ ε (x ε ) − αρ l (x ε )| > C. By Lemma 1 of [22], for sufficiently smallσ > 0, for any x ∈ [h l −σ, h l +σ] := Iσ and x ε ∈ Iσ we have (3.17) |αρ l (x ε ) − αρ l (x)| < 2C/5.
We are now ready to prove Theorem 3.4.
Proof. (of Theorem 3.4) Let ρ 0 = αρ l + (1 − α)ρ r be the limit of ρε obtained in (1) of Lemma 3.14. We will show that α/(1 − α) = l.a.h.r. This will imply the first part of Theorem 3.4. We first observe that   Similarly we can obtain

discussion
In this paper we studied both open and metastable systems. In Theorem 2.1, we rigorously derived first-order asymptotic escape rate formulae for piecewise expanding maps of the interval with random holes. In [27] the authors studied a deterministic sequence of nested holes and derived a first-order asymptotic escape rate formulae for piecewise expanding maps of the interval with holes. In this paper the sequence of holes is random. Moreover, the random holes we consider are not necessarily nested. Consequently our work generalizes the pioneering work of [27] to the random setting. It would be interesting to explore whether one can obtain higher order approximations in the random setting, even for a simpler class of maps, such as circle maps. A study for higher order approximations of escape rates for deterministic holes using circle maps was carried in [14]. Other possible generalizations, such as higher dimensional systems, were discussed earlier in the introduction.
In the second part of the paper we studied the problem of approximating the unique stationary density for a random system which initially contains two separate ergodic components. We have shown in Theorem 3.4 that the unique stationary density of the random system can be approximated by a weighted combination of the two initially separate ergodic components of the deterministic system. Furthermore, we have shown that the weights are given by the ratio of the escape rates on the individual ergodic components. A deterministic perturbation version of the above result was first studied in [22] for piecewise expanding maps. A version of [22] for intermittent maps was obtained in [3]. It is worth restating (see introduction) that it would be also interesting to obtain a random version in the case of intermittent maps.