Wael Bahsoun
email: w.bahsoun@lboro.ac.uk

Sandro Vaienti
email: vaienti@cpt.univ-mrs.fr

ESCAPE RATES FORMULAE AND METASTABLILTY FOR RANDOMLY PERTURBED MAPS

Keywords: 1991 Mathematics Subject Classification. Primary 37A05, 37E05 Escape Rates, Metastability, Random perturbations, Expanding maps, Invariant Densities. We would like

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 [START_REF] Demers | Escape rates and conditionally invariant measures[END_REF][START_REF] Dettmann | Recent advances in open billiards with some open problems[END_REF]. 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 [START_REF] Bunimovich | Where to place a hole to achieve a maximal escape rate[END_REF] 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 [START_REF] Keller | Rare events, escape rates, quasistationarity: some exact formulae[END_REF] 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 [START_REF] Bunimovich | Where to place a hole to achieve a maximal escape rate[END_REF][START_REF] Keller | Rare events, escape rates, quasistationarity: some exact formulae[END_REF], other researchers studied this phenomenon for different types of systems [START_REF] Afraimovich | Which hole is leaking the most: a topological approach to study open systems[END_REF][START_REF] Demers | Behavior of the Escape Rate Function in Hyperbolic Dynamical Systems[END_REF][START_REF] Ferguson | Escape rates for Gibbs measures[END_REF] 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 [START_REF] González Tokman | Approximating invariant densities of metastable systems[END_REF][START_REF] Dolgopyat | The Diffusion Coefficient for Piecewise Expanding Maps of the Interval with Metastable States[END_REF][START_REF] Froyland | Spectral degeneracy and escape dynamics for intermittent maps with a hole[END_REF][START_REF] Keller | Rare events, escape rates, quasistationarity: some exact formulae[END_REF]. 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 [START_REF] Gaspard | Heat Conduction and Fouriers Law by Consecutive Local Mixing and Thermalization[END_REF]. They also play an important in astronomy [START_REF] Waalkens | Escape from planetary neighbourhoods[END_REF]. In molecular dynamics, almost invariant regions of metastable systems are used to identify sets where stable molecular conformations occur [START_REF] Schütte | Transfer operator approach to conformational dynamics in biomolecular systems[END_REF]. In astrodynamics, metastable systems are used to recognize regions from which asteroids escape is infrequent [START_REF] Dellnitz | Transport in dynamical astronomy and multibody problems[END_REF]. Moreover, metastable dynamical systems have been recently used to develop realistic models of atmospheric and ocean circulation [START_REF] Dellnitz | Seasonal variability of the subpolar gyres in the Southern Ocean: a numerical investigation based on transfer operators[END_REF][START_REF] Froyland | Detecting coherent oceanic structures via transfer operator[END_REF][START_REF] Santitissadeekorn | Optimally coherent sets in geophysical flows: A new approach to delimiting the stratospheric polar vortex[END_REF]. 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 [START_REF] Collet | On the enhancement of diffusion by chaos, escape rates and stochastic instability[END_REF]. Our result is in the spirt of the recent trend of describing the escape rate as a function of the position of the hole [START_REF] Afraimovich | Which hole is leaking the most: a topological approach to study open systems[END_REF][START_REF] Bunimovich | Where to place a hole to achieve a maximal escape rate[END_REF][START_REF] Ferguson | Escape rates for Gibbs measures[END_REF][START_REF] Keller | Rare events, escape rates, quasistationarity: some exact formulae[END_REF]. In particular, our result generalizes Keller-Liverani escape rates formulae [START_REF] Keller | Rare events, escape rates, quasistationarity: some exact formulae[END_REF] 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 [START_REF] González Tokman | Approximating invariant densities of metastable systems[END_REF]. 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 [START_REF] Bahsoun | Metastability of certain intermittent maps[END_REF]. 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 [START_REF] Alves | Strong stochastic stability for non-uniformly expanding maps[END_REF][START_REF] Shen | On stochastic stability of non-uniformly expanding interval maps[END_REF] 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.

2.

Escape Rate Formulae For Randomly Perturbed Maps 2.1. Notation. Throughout the paper we use the following notation: (I, B, m) is the measure space where I = [0, 1], B is the Borel σ-algebra and m is Lebesgue measure. For f ∈ L 1 (I, B, m), we define

V f = inf{var f : f = f a.e.}, where var f = sup{ t-1 j=0 |f (x j+1 ) -f (x j )| : 0 = x 0 < x 1 < • • • < x t = 1}.
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 {I i } q i=1 , I i ∩ I j = ∅ for i = j, Īi := [c i,0 , c i+1,0 ] and there exists δ > 0 such that T i,0

:= T | (ci,0,ci+1,0) is C 2 which extends to a C 2 function Ti,0 on a neighbourhood [c i,0 -δ, c i+1,0 + δ] of Īi ; (O2) inf x∈I\C0 |T (x)| ≥ κ -1 > 1, where C 0 = {c i,0 } q i=1 . (O3)
T preserves a unique acim3 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. 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

X ω := I \ H ω ,
and define for f ∈ L 1 (I, B, m)

(2.1) Lε f (x) := Ωε L 0 (f 1 Xω )(x)dθ ε ,
where L 0 is the transfer operator (Perron-Frobenius) [START_REF] Baladi | Positive transfer operators and decay of correlations[END_REF][START_REF] Boyarsky | Laws of Chaos[END_REF] associated with T ; i.e., for f ∈ L 1 (I, B, m) we have

L 0 f (x) = y:=T -1 (x) f (y) |T (y)| .
The transfer operator Lε 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

1 A L 0 f = L 0 (1 T -1 A f ), we obtain Ln ε f (x) = Ωε • • • Ωε L n 0 (f 1 Xω 1 ∩T -1 Xω 2 ∩•••∩T -(n-1) Xω n )(x)dθ ε (ω 1 ) • • • dθ ε (ω n ) def := Ωε L n 0 (f 1 Xω 1 ∩T -1 Xω 2 ∩•••∩T -(n-1) Xω n )(x)dθ ∞ ε (ω). (2.2)
where θ ∞ ε := Π ∞ i=1 θ ε and ω := (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 [START_REF] Keller | Rare events, escape rates, quasistationarity: some exact formulae[END_REF].

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 Lε g ε = e ε g ε .

Moreover,

(1) If z is a non-periodic point of T , then

lim ε→0 1 -e ε Ωε µ(H ω )dθ ε (ω) = 1.
(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 (C) θ {ω; m(H ω ) ∈ ( -υ , )} > 1 -υ ; then lim ε→0 1 -e ε Ωε µ(H ω )dθ ε (ω) = 1 - 1 |(T p ) (z)| . 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 [START_REF] Keller | Stability of the spectrum for transfer operators[END_REF] in the proof our Lemma 2.7. The strict positivity of the density g ε also follows from [START_REF] Keller | Stability of the spectrum for transfer operators[END_REF] since µ is equivalent to m.

Remark 2.3. Since Lε g ε = e ε g ε , with I g ε dm = 1, by using the definition of Lε , it follows that:

e ε = Ωε dθ ε (ω) I g ε 1 Xω dm.
Set ν ε to be the Borel probability measure: 4 It is very easy to construct examples that satisfy condition (C). For instance, suppose we associate to any ω ∈ Ωε a symmetric hole around the point z. Then:

ν ε (A) def := 1 e ε I Ωε 1 A 1 Xω g ε dθ ε dm,
(i) Hε = [z -ε 2 ; z + ε 2 ]; Hω = [z -ω 2 ; z + ω 2 ]; (ii) m(Hε) = ε; m(Hω) = ω.
Let us choose an absolutely continuous θε with density dε. Our condition (C) will be satisfied whenever ε ε-ε υ dε(ω)dω > 1 -ε υ which can be obtained, for instance, by taking dε = ε -υ on the interval (ε -ε υ , ε) and 0 everywhere else.

where A ⊆ I is a measurable set. By using the fact that Lε g ε = e ε g ε and the definition of Lε , we obtain that ν ε satisfies the following:

(2.3) ν ε (A) = 1 e ε Ωε dθ (ω)ν (T -1 A ∩ T -1 X ω ); (2.4) e ε = Ωε dθ ε (ω)ν ε (T -1 X ω ); and (2.5) e n ε ν ε (A) = Ωε dθ ∞ ε (ω)ν ε (T -n A ∩ T -1 X ω1 ∩ T -2 X ω2 ∩ • • • ∩ T -n X ωn ).
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 Lε . This operator enjoys all the properties of what Keller [START_REF] Keller | Rare events, exponential hitting times and extremal indices via spectral perturbation[END_REF] 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 for Lε and L 0 . A deterministic version of Lemma 2.6 can be found in [START_REF] Liverani | Lasota-Yorke maps with holes: conditionally invariant probability measures and invariant probability measures on the survivor set[END_REF].

Lemma 2.6. There exists a γ ∈ (0, 1) and constants A, B > 0 such that for any n ≥ 1 and f ∈ BV (I) we have

||L n 0 f || BV ≤ Aγ n ||f || BV + B||f || 1 ; || Ln ε f || BV ≤ Aγ n ||f || BV + B||f || 1 .
Proof. We begin to compute the total variation, leaving the L 1 estimate at the end.

Let Z n = Z ∨ T -1 Z • • • ∨ T -(n-1) Z, where Z = {I i } q i=1 . Let J n = 1 |(T n ) | . For A ∈ Z n and f ∈ BV (I): there is a κ ∈ (0, 1), such that (2.6) V 1 T n A (J n f ) • T -n |A ≤ 2κ n V A f + 1 min A∈Z n m(A) A |f |dm.
In particular,

(2.7) V I L n 0 (f ) ≤ A∈Z n V 1 T n A (J n f ) • T -n |A ≤ 2κ n V I f + 1 min A∈Z n m(A) I |f |dm.
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

X ω1 ∩ T -1 X ω2 ∩ • • • ∩ T -(n-1) X ωn ∩ A consists of at most n + 1 connected components, we obtain V 1 T n A (J n f 1 Xω 1 ∩T -1 Xω 2 ∩•••∩T -(n-1) Xω n ) • T -n |A = V 1 T n A (J n g) • T -n |A ≤ 2κ n V A g + 1 min A∈Z n m(A) A |g|dm ≤ 2κ n (V A f + 2(n + 1) sup x∈A |f |) + 1 min A∈Z n m(A) A |f |dm ≤ 2κ n (2n + 3)V A f + 1 min A∈Z n m(A) (2κ n (2n + 2) + 1) A |f |dm.
(2.8)

Summing over A ∈ Z n in (2.8) we obtain

V I L n 0 (f 1 Xω 1 ∩T -1 Xω 2 ∩•••∩T -(n-1) Xω n ) ≤ 2κ n (2n + 3)V I f + 1 min A∈Z n m(A) (2κ n (2n + 2) + 1) I |f |dm (2.9)
Since the inequality in (2.9) does not depend on ω and

Ln ε f (x) = Ωε • • • Ωε L n 0 (f 1 Xω 1 ∩T -1 Xω 2 ∩•••∩T -(n-1) Xω n )(x)dθ ε (ω 1 ) • • • dθ ε (ω n ),
we also have

(2.10) V I Ln ε f ≤ 2κ n (2n + 3)V I f + 1 min A∈Z n m(A) (2κ n (2n + 2) + 1) I |f |dm.
The estimate on || Ln 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

|| Ln ε f || 1 ≤ |f |1 Xω 1 ∩T -1 Xω 2 ∩•••∩T -(n-1) Xω n )dm ≤ ||f || 1 .
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 for Lε and L 0 . Lemma 2.7. Consider Lε : BV (I) → BV (I). Then there are e ε ∈ (0, 1), ϕ ε ∈ BV (I), a probability Borel measure ν ε and linear operators

Q ε : BV (I) → BV (I) such that (1) e -1 ε Lε = ϕ ε ⊗ ν ε + Q ε ; (2) Lε ϕ ε = e ε ϕ ε , ν ε Lε = e ε ν ε , Q ε ϕ ε = 0, ν ε Q ε = 0; (3) ∞ n=0 sup ε ||Q n ε || BV (I) < ∞; (4) m(ϕ ε ) = 1 and sup ε ||ϕ ε || BV (I) < ∞; (5) ∃ C > 0 such that η ε := sup ||ψ|| BV (I) ≤1 | I (L 0 -Lε )ψdm| → 0 as ε → 0, and η ε • ||(L 0 -Lε )ρ|| BV (I) ≤ C|∆ ε |,
where ∆ ε := m((L 0 -Lε )ρ).

Proof. We have for any ψ ∈ BV (I):

(2.11) (L 0 -Lε )ψ = Ωε L 0 (ψ1 Hω )dθ ε .
Since ψ is also in L ∞ (m) we have (2.12)

η ε = sup ||ψ|| BV (I) ≤1 | I (L 0 -Lε )ψdm| ≤ Ωε m(H ω )dθ ε ≤ m(H ε ) → 0 as ε → 0.
Thus, by Lemma 2.6 and (2.12), using the abstract perturbation result on the stability of spectrum of transfer operators [START_REF] Keller | Stability of the spectrum for transfer operators[END_REF], we obtain ( 1)-( 4) of the lemma. Now by (2.11), notice that

(2.13) η ε ≤ A ε := Ωε m(H ω )dθ ε , and
(2.14) ∆ ε = Ωε µ(H ω )dθ ε .
Moreover, using (2.6), condition (O3), and calling:

(i) K = max(Aγ, B); (ii) C = 3||ρ|| BV ; (iii) ρ m := inf x∈I ρ(x),
x -m a.e., we obtain

||(L 0 -Lε )ρ|| BV (I) = || Ωε L 0 (ρ1 Hω )dθ ε || BV (I) (2.15) ≤ K Ωε (V (ρ1 Hω ) + ||ρ1 Hω || 1 ) dθ ε ≤ K C ρ m Ωε µ(H ω )dθ ε A ε .
Thus, by (2.13), (2.14) and (2.15),

η ε • ||(L 0 -Lε )ρ|| BV (I) ≤ K C ρ m • ∆ ε .
We are now ready to apply the abstract perturbation result of [START_REF] Keller | Rare events, escape rates, quasistationarity: some exact formulae[END_REF]. Indeed, Lemma 2.7 shows that all the conditions imposed in [START_REF] Keller | Rare events, escape rates, quasistationarity: some exact formulae[END_REF] are satisfied by our random systems.

Lemma 2.8.

(1) For sufficiently small ε, ∆ ε = 0.

(2) If for each integer k ≥ 0 the following limit:

q k := lim ε→0 q k,ε := lim ε→0 m((L 0 -Lε ) Lk ε (L 0 -Lε )(ρ)) ∆ ε exists, then lim ε→0 1 -e ε ∆ ε = 1 - ∞ k=0 q k .
Proof. By condition (O3), for sufficiently small ε > 0, ∆ ε > 0. Thus, Part (2) of the lemma follows from the abstract perturbation result of [START_REF] Keller | Rare events, escape rates, quasistationarity: some exact formulae[END_REF], 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 have

1 A L 0 f = L 0 (1 T -1 A f ).
Consequently, by (2.11) and (2.2), we obtain

m((L 0 -Lε ) Lk ε (L 0 -Lε )(ρ)) = Ωε I ((L 0 -Lε ) Lk ε (L 0 -Lε )(ρ))dmdθ ε (ω) = Ωε I L 0 (1 Hω • Lk ε (L 0 -Lε )(ρ))dmdθ ε (ω) = Ωε I 1 Hω • Lk ε (L 0 -Lε )(ρ))dmdθ ε (ω) = Ωε I Ωε 1 Hω • L k 0 1 Xω 1 ∩•••∩T -(k-1) Xω k • (L 0 -Lε )(ρ))dθ ε (ω)dmdθ ε (ω) = Ωε I Ωε L k 0 (1 T -k Hω • 1 Xω 1 ∩•••∩T -(k-1) Xω k • (L 0 -Lε )(ρ))dθ ε (ω)dmdθ ε (ω) = Ωε Ωε I 1 T -k Hω • 1 Xω 1 ∩•••∩T -(k-1) Xω k • (L 0 -Lε )(ρ))dmdθ ε (ω)dθ ε (ω) = Ωε Ωε I Ωε 1 T -k Hω • 1 Xω 1 ∩•••∩T -(k-1) Xω k • (L 0 1 H ω ρ)dmdθ ε (ω)dθ ε (ω)dθ ε (ω ) = Ωε Ωε I Ωε L 0 (1 T -(k+1) Hω • 1 T -1 Xω 1 ∩•••∩T -k Xω k • 1 H ω ρ)dmdθ ε (ω)dθ ε (ω)dθ ε (ω ) = Ωε Ωε Ωε µ [T -(k+1) H ω ∩ T -1 X ω1 ∩ • • • ∩ T -k X ω k ] ∩ H ω dθ ε (ω )dθ ε (ω)dθ ε (ω).
(2.16)

Now, if z is not a periodic point of T , since T is continuous at z, for sufficiently small ε > 0,

[T -1 X ω1 ∩ • • • ∩ T -k X ω k ∩ T -(k+1) H ω ] ∩ H ω = ∅.
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

(2.17) m(T -p H ω ∩ H ω )dθ (ω)dθ (ω ) m(H ω )dθ (ω) .
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

m(T -p H ω ∩ H ω ) = Hω,p∩H ω dm = T p (Hω,p∩H ω ) |DT p (y)| -1 dm(y) = Hω∩T p H ω |DT p (y)| -1 dm(y).
(2.18) Equation (2.18) gives immediately the upper bound

(2.19) m(T -p H ω ∩ H ω ) ≤ sup Hω |DT p | -1 m(H ω ) ≤ sup Hε |DT p | -1 m(H ω ).
By (2.19) and the continuity of DT p at z we obtain

(2.20) lim ε→0 m(T -p H ω ∩ H ω )dθ (ω)dθ (ω ) m(H ω )dθ (ω) ≤ |DT p | -1 (z).
For the lower bound, since T p is expanding, we have

m(T -p H ω ∩ H ω ) ≥ Hω∩H ω |DT p (y)| -1 dm(y) = Hω |DT p (y)| -1 dm(y) - Hω/H ω |DT p (y)| -1 dm(y) ≥ inf Hε |DT p | -1 m(H ω ) - Hω/H ω |DT p (y)| -1 dm(y). (2.21)
Therefore,

m(T -p H ω ∩ H ω )dθ (ω)dθ (ω ) m(H ω )dθ (ω) ≥ inf Hε |DT p | -1 dθ (ω)dθ (ω ) - Hω/H ω |DT p (y)| -1 dm(y)dθ (ω)dθ (ω ) m(H ω )dθ (ω) . (2.22)
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

Hω/H ω |DT p (y)| -1 dm(y) ≤ sup Hω |DT p | -1 m(H ω ∆H ω ). Now, let us call G ω the complement of H ω in H ε ; we immediately have by the condition (C): m(H ω ∩ H ω ) = m(H ω ) -m(H ω ∩ G ω ) ≥ (ε -ε υ ) -ε υ ≥ ε - 2ε υ . This implies that m(H ω /H ω ) = m(H ω ) -m(H ω ∩ H ω ) ≤ 2ε υ and therefore m(H ω ∆H ω ) ≤ 4ε υ . Let us call F ε := {ω ∈ Ω ε ; m(H ω ) ∈ (ε -ε υ , ε)}; Q ε := {(ω, ω ) ∈ Ω ε × Ω ε ; m(H ω ∆H ω )} ≤ 4ε υ . Notice that Q ε ⊃ F ε × F ε and recall that θ ε (F ε ) ≥ 1 -ε υ .
Then

Hω/H ω |DT p (y)| -1 dm(y)dθ (ω)dθ (ω ) m(H ω )dθ (ω) ≤ m(H ω ∆H ω )dθ (ω)dθ (ω ) m(H ω )dθ (ω) ≤ 1 m(H ω )dθ (ω) [ Q m(H ω ∆H ω )dθ (ω)dθ (ω ) + Q c ε m(H ω ∆H ω )dθ (ω)dθ (ω )] ≤ 1 m(H ω )dθ (ω) [4 υ + θ 2 ε (F c ε ))] ≤ 1 Fε m(H ω )dθ (ω) [4 υ + ε 2υ ] ≤ 4 υ + ε 2υ ( -υ )(1 -υ )
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

(2.23) lim ε→0 m(T -p H ω ∩ H ω )dθ (ω)dθ (ω ) m(H ω )dθ (ω) ≥ |DT p | -1 (z).
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 [START_REF] González Tokman | Approximating invariant densities of metastable systems[END_REF]. Deterministic intermittent systems of this type were studied in [START_REF] Bahsoun | Metastability of certain intermittent maps[END_REF]. Other recent results on metastable random dynamical systems have been obtained in [START_REF] Froyland | Metastability, Lyapunov exponents, escape rates, and topological entropy in random dynamical systems[END_REF][START_REF] Góra | Metastable systems as random maps[END_REF]. 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 {I i } q i=1 , I i ∩ I j = ∅ for i = j, Īi := [c i,0 , c i+1,0 ] and there exists δ > 0 such that T i,0 (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 [START_REF] González Tokman | Approximating invariant densities of metastable systems[END_REF], 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 [START_REF] González Tokman | Approximating invariant densities of metastable systems[END_REF]. 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.

:= T | (ci,0,ci+1,0) is C 2 which extends to a C 2 function Ti,0 on a neighbourhood [c i,0 -δ, c i+1,0 + δ] of Īi ; (A2) inf x∈I\C0 |T (x)| ≥ β -1 0 > 2, where C 0 = {c i,0 } q i=1 . (
An example of a map T 0 satisfying the above assumption is shown in Figure 2. We denote the transfer operator (Perron-Frobenius) [START_REF] Baladi | Positive transfer operators and decay of correlations[END_REF][START_REF] Boyarsky | Laws of Chaos[END_REF] 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 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.

ω n = (ω 1 , ω 2 , • • • , ω n ),
U * ε Λ(A) = I Ωε 1 A (T ω (x))dθ ε (ω)dΛ(x) = I U ε 1 A (x)dµ ε (x),
where U ε is the random evolution operator acting on L ∞ m functions g:

(3.1) U ε g = Ωε g • T ω dθ ε (ω).
A measure µ ε on (I, B(I)) is called a T ε -stationary measure if and only if, for any A ∈ B(I),

(3.2) U * ε µ ε (A) = µ ε (A).
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:

(3.3) (L ε f )(x) = Ωε L ω f (x)dθ ε (ω),
which satisfies the duality condition (3.4)

I gL ε f dm = I U ε gf dm
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, Īi,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):

[c i,0 + δ, c i+1,0 -δ] ⊂ [c i,ω , c i+1,ω ] ⊂ [c i,0 -δ, c i+1,0 +
δ]; 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 :

C ω = {c i,ω } q i=1 .
The last condition, as explained in section 2.4 of [START_REF] González Tokman | Approximating invariant densities of metastable systems[END_REF] 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 [START_REF] Baladi | On the spectra of randomly perturbed expanding maps[END_REF]) 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

(3.7) 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 [START_REF] Góra | Graph-theoretic bound on number of A.C.I.M. for random transformation[END_REF]. 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 l,ω := I l ∩ T -1 ω (I r ) and 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 ) (3.8) Ll,ε f (x) := Ωε L l,0 (f 1 X l,ω )(x)dθ ε ,
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 operator Lr,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: (2) Moreover,

lim ε→0 1 -e l,ε 1 -e r,ε = α 1 -α .
Remark 3.6. As it was observed in [START_REF] González Tokman | Approximating invariant densities of metastable systems[END_REF], 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.

Lemma 3.7. We have

Ωε µ ε (H l,ω )dθ ε (ω) = Ωε µ ε (H r,ω )dθ ε (ω)
, where dµ ε = ρ ε dm.

Proof. We have

(3.10) µ ε (I l ) = Ωε µ ε (I l )dθ ε (ω) = Ωε µ ε (X l,ω )dθ ε (ω) + Ωε µ ε (H l,ω )dθ ε (ω).
Since T -1 ω (I l ) = X l,ω ∪ H r,ω , we also have

(3.11) Ωε µ ε (T -1 ω I l )dθ ε (ω) = Ωε µ ε (X l,ω )dθ ε (ω) + Ωε µ ε (H r,ω )dθ ε .
Using (3.10), (3.11) and that µ ε is T ε -invariant the lemma follows.

Lemma 3.8. There exists a constant K, depending only on the map T , such that for 0 ≤ n < ∞ there exists σ, depending on n, and ε σ , depending on σ, such that ∀ε ≤ ε σ and σ ≤ σ, we have

(1) L n ε 1 is a C 1 -function on I σ = [h * -σ, h * + σ], * ∈ {l, r}; (2
) for any x, y ∈ I σ , we have

|L n ε 1(x) -L n ε 1(y)| ≤ K • |x -y|; in particular, |L n ε 1(x) -L n ε 1(y)| ≤ K • σ. Proof.
By condition (A4) we have for every n > 0, (T n 0 C 0 ) ∩ H 0 = ∅. In particular, for a fixed n ∈ N, we can choose σ > 0 so small so that, for 0 ≤ k ≤ n, (T k 0 C 0 ) ∩ (h * -σ , h * + σ ) = ∅. We now prove that there is a possibly smaller subinterval (h * -σ, h * + σ), σ ≤ σ upon which (1) and ( 2) of the lemma are satisfied. Let us give the explicit form of this operator.

The iterates of L ε are given by:

(L n ε 1) = • • • L ω1 (L ω2 (• • • L ωn (1))) dθ ε (ω 1 ) • • • dθ ε (ω n ).
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:

Ψ ωn,••• ,ω1 (x) := L ωn L ωn-1 (• • • L ω1 (1)) (x) = k1,...,kn 1 |D(T ωn • • • • • T ω1 )((T -1 k1,ω1 • • • • • T -1 kn,ωn )(x))| × 1 Ω k 1 ,••• ,kn ω 1 ,••• ,ωn ) (x).
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 = T ωn I kn,ωn ∩ T ωn T ωn-1 I k1,ω k-1 ∩ • • • ∩ T ωn T ωn-1 • • • • • T ω1 I k1,ω1
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 ) -Tl,0 (c l,ω )| + | Tl,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 ) -Tl,0 (c l,ω )| ≤ sup l, x∈I l,0 |D Tl,0 ||c l,0 -c l,ω | ≤ σ /2 and | Tl,0 (c l,ω ) -T l,ω (c l,ω )| ≤ σ /

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

Ψ ωn+1,••• ,ω1 = L ωn+1 Ψ ωn,••• ,ω1 .
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)| ≤ (3.12) k1,...,kn 1 |D(T ωn • • • • • T ω1 )(T -1 ωn (x))| - 1 |D(T ωn • • • • • T ω1 )(T -1 ωn (y))|
where we skip the characteristic function since both x, y will be in the same Ω k1,••• ,kn ω1,••• ,ωn . We have

D 1 |D(T ωn • • • • • T ω1 )(z) = n-1 k=0 D 2 T ω n-k n-1-k l=1 T ω n-l z DT ω n-k n-1-k l=1 T ω n-l z 2 k j=0 DT ωn-j+1 n-j l=1 T ω n-l z . (3.13) Since sup ω,x∈I k,ω D 2 T ω (x) DT ω (x) ≤ C 1 < ∞,
which follows from our assumptions on the map T and its perturbations, and

inf ω,x∈I k,ω |DT ω (x)| ≥ β > 2,
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

(3.12) ≤ C 2 k1,••• ,kn |T -1 ωn (x)-T -1 ωn (y)| ≤ |x-y| k1,••• ,kn 1 |D(T ω1 • • • • • T ωn )(T -1 ωn (ζ))|
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) (3.12) ≤ C 3 |x -y| k1,••• ,kn 1 |D(T ω1 • • • • • T ωn )(T -1 ωn (x))|
.

We now integrate (3.14) over the noise and get, for any x, y ∈ I σ ,

|L n ε 1(x) -L n ε 1(y)| ≤ C 3 σL n ε 1 ≤ C 3 σ(1 + B)
, where L n ε 1 ≤ 1 + B is obtained from the Lasota-Yorke inequality in Proposition 3.2. The Lemma finally follows by choosing K = C 3 (1 + B).

In [START_REF] González Tokman | Approximating invariant densities of metastable systems[END_REF], the authors dealt with the deterministic version of this section of our paper. In [START_REF] González Tokman | Approximating invariant densities of metastable systems[END_REF], 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 variation6 . 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

f sal = u∈S s u H u ,
where S is a countable set, and

H u (x) =    -1 if x < u -1 2 if x = u 0 if x > u .
This representation imposes the boundary condition f sal (1) = 0. Moreover, the variation of the singular part satisfies

V (f sal ) = u∈S |s u | ≤ V (f ). Call F n,ε := 1 n n-1 k=0 L k ε 1 := 1 n n-1 k=0 Ωε L k ω 1dθ ∞ ε (ω),
where

L k ω = L ω1 • • • L ω k .
Note that, by (R1), we have lim

n→∞ ||F n,ε -ρ ε || 1 = 0.
We have already shown in Lemma 3.8 that the set of discontinuities of L n ω is given by S n,ω = ∪ n-1 j=0 ∪ kn-j T ωn T ωn-1 • • • T ωn-j ∂I kn-j ,ωn-j where k n-j runs over the domains of injectivity of T ωn-j and ∂I denotes the endpoint of the connected interval I. We rewrite this set as S n,ω = ∪ n-1 j=0 Ŝj,ω , where Ŝj,ω := ∪ kn-j T ωn T ωn-1 • • • T ωn-j ∂I kn-j ,ωn-j . We also write Sn := ∪ ω S n,ω and Ŝj = ∪ ω Ŝj,ω .

Notice that Ŝj ⊂ Sn . Finally, we write S := ∪ n≥1 Sn .

Lemma 3.9.

(1) Sn is a finite set and S is a countable set.

( 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] \ Sn 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 to Sn .

Proof. We prove the lemma by contradiction. Namely, let a be a discontinuity point for F n,ε , and suppose that a / ∈ Sn ,. Then ∀ω and 0 ≤ k ≤ n -1, we have that

(3.15) lim x→a L k ω 1(x) = L k ω 1(a).
Since, for sufficiently small , each L ω , ω ∈ Ω ε , satisfies a Lasota-Yorke inequality with uniform constants (see Proposition 3.2)

V L ω f ≤ ηV f + B 0 ||f || 1 ,
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

F n,ε (a) = 1 n n-1 k=0 Ωε L k ω 1(a) = 1 n n-1 k=0 Ωε lim x→a L k ω 1(x) = lim x→a 1 n n-1 k=0 Ωε L k ω 1(x) = lim x→a F n,ε (x).
Thus, a contradiction.

Let #u = inf{j ≥ 1; u ∈ Ŝj }.
Lemma 3.12.

(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. 7 One can easily construct a random system with such a bad behaviour. For example, let us write the formula of the first branch for the map T 0 in Figure 2. We have T 0 (x) := 3x for 0 ≤ x ≤ 1/6. For a fixed rational number ε, Let Ωε = Q ∩ [0, ε]. Then, each ω ∈ Ωε is a rational number. Further, define the first branch of Tω(x) := (3 + 6ω)x, for 0 ≤ x ≤ 1/6. Then ω + 1/2 is in the postcritical set of Tω. In particular S1 ∩ [1/2, 1/2 + ε] = {ω + 1/2} ω∈Ωε . Consequently, [0, 1] \ ( S1 ∩ [1/2, 1/2 + ε]) is an uncountable set and it does not contain any interval.

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 in Sn ⊂ 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 |s u,n | ≤ V (F n,ε ) ≤ V (L k ε 1) ≤ C .
Moreover,

F n,ε = n -1 n L ε F n-1,ε + 1 n .
Now for #u > 1, i.e. u / ∈ Ŝ1 , 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 |s u,n | ≤ n -1 n 1 η v∈S;T ω v=u |s v,n-1 |dθ ε (ω ). Then if 0 < m < n we have #u>m |s u,n | ≤ #u>m n -1 n 1 η v∈S;T ω v=u |s v,n-1 |dθ ε (ω ) ≤ n -1 n 1 η #u>m-1 |s u,n-1 | ≤ • • • ≤ n -m n 1 η m #u>0 |s u,n-m | ≤ 1 η m C .
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

F sal ε := u∈S s u H u . 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] \ Sn . Therefore, on [0, 1] \ Sn , 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] \ Sn ). 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] \ Sn . By (2) Lemma 3.9, [0, 1] \ Sn 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 .

Lemma 3.13.

(1) For any τ > 0 there exist an n, 0 ≤ n < ∞, a σ depending on n, and an ε σ , depending on σ, such that ∀ε ≤ ε σ and σ ≤ σ, we have

V Iσ ρ sal ε < η -n C < τ, where I σ = [h * -σ, h * + σ], * ∈ {l,
r} and C is the constant which appears in Lemma 3.12.

(2) For any x, y ∈ I σ , there exists a uniform constant K such that

|ρ reg ε (x) -ρ reg ε (y)| ≤ K • |x -y|. Proof.
Choose n large enough so that η -n C < τ . Then, using (1) of Lemma 3.8, for n > 0 such that η -n C < τ we can find a σ, depending on n, and an ε σ , depending on σ, such that ∀ε ≤ ε σ and σ ≤ σ, L n ε 1 is a C 1 -function on I σ . Consequently, using Lemma 3.12 with m = n, we obtain part (1) of the lemma. Part (2) of the lemma is a consequence of part (2) of Lemma 3.12.

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 values ε ∈ Ω ε for which ρ ε 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 ρ ε with ε ∈ Ω ε ; then there exists 0 ≤ α ≤ 1 such that: 8(1) Proof. We argued above that (1) is true. The proof of (3) will be identical to that of (2). Thus, we only prove [START_REF] Alves | Strong stochastic stability for non-uniformly expanding maps[END_REF]. Observe that for all ω ∈ Ω ε 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 of ε 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 [START_REF] González Tokman | Approximating invariant densities of metastable systems[END_REF], for sufficiently small σ > 0, for any x ∈ [h l -σ, h l + σ] := I σ and x ε ∈ I σ we have (3.17) |αρ l (x ε ) -αρ l (x)| < 2C/5.

ρ 0 = αρ l + (1 -α)ρ r ; (2)
Choose n big enough so that C η -n < C/5. By Lemma 3.13 we can find a σ depending on n, and an ε σ , depending on σ, such that ∀ε ≤ ε σ and σ ≤ σ, we have

V Iσ ρ sal ε < C/5,
and for any x, y ∈ I σ , there exists a uniform constant K such that

|ρ reg ε (x) -ρ reg ε (y)| ≤ K • |x -y| ≤ K σ.
We make sure to take σ small enough such that K σ < C/5, and σ ≤ σ. Then for x, x ε ∈ I σ we have 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 ρ ε 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 (3.20)

Let h l be the infinitesimal hole in I l . By condition (A4) ρ 0 is continuous on h l . This implies lim ε→0 µ l (H l ,ω) m(H l ,ω) = ρ l (h l ) > 0 by condition (A5). Therefore, by using (2) Lemma 3.14 and (3.20), we obtain 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 [START_REF] Keller | Rare events, escape rates, quasistationarity: some exact formulae[END_REF] 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 [START_REF] Keller | Rare events, escape rates, quasistationarity: some exact formulae[END_REF] 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 [START_REF] Dettmann | Open circle maps: Small hole asymptotics[END_REF]. 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 [START_REF] González Tokman | Approximating invariant densities of metastable systems[END_REF] for piecewise expanding maps. A version of [START_REF] González Tokman | Approximating invariant densities of metastable systems[END_REF] for intermittent maps was obtained in [START_REF] Bahsoun | Metastability of certain intermittent maps[END_REF]. It is worth restating (see introduction) that it would be also interesting to obtain a random version in the case of intermittent maps.

Figure 1 .

 1 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.

  A3) ∃ b in the interior of I such that T | I * ⊆ I * , where * ∈ {l, r}, I * is an interval such that I l ∪ I r = I and I l ∩ I r = {b}. (A4) Let H 0 := T -1 {b} \ {b}. We call H 0 the set of infinitesimal holes and we assume that for every n ≥ 1, (T n C 0 ) ∩ H 0 = ∅.

  defined as : T ω n := T ωn • • • • • T ω1 (x) (random transformations). This defines a Markov process T ε with transition functionP(x, A) = Ωε 1 A (T ω (x))dθ ε (ω),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:

  0 -c i,ω | = 0 (uniform collapsing of boundary points) (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 (3.6) lim ε→0 | Ti,0 (x) -T i,ω (x)| = 0 (uniform convergence of maps) (B2) Boundary condition: (i) if b / ∈ C 0 , then T (b) = b and for all ω > 0, T ω (b) = b; (ii) if b ∈ C 0 , then T (b-) < b < T (b+) and for all ω > 0, b ∈ C ω , where

Figure 3 .

 3 Figure 3. A typical graph of one of the constituent maps T ω of the metastable random map T ε corresponding to the initial system shown in Figure2.

Theorem 3 . 4 .Corollary 3 . 5 . 1 )

 34351 Let ρ ε be the unique T ε -invariant density. Suppose the l.a.h.r exists then limε→0 ||ρ ε -ρ 0 || 1 = 0,where ρ 0 = αρ l + (1 -α)ρ r and l.a.h.r = α 1-α .Theorem 2.1 will give us additional information about the ratio α 1-α which appears in Theorem 3.4 under the additional assumption:(B4) The map T is continuous on h * and ∀ω, h * ∈ H * ,ω . Moreover, (I * , µ * , T * ), * ∈ {l, r}, is mixing. Assume condition (B4). Then (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 .

3 )

 3 ) [0, 1] \ Sn is a finite collection of open intervals. (For any ω := (ω n , . . . , ω 1 ), T ω is C 2 on each open interval belonging to [0, 1] \ Sn . Proof. Since Ω ε is a finite set, (1) follows by definition of Sn and S. (2) is a consequence of (1). (3) follows by definition of Sn .

  |ρ ε(x) -(1 -α)ρ r (x)| = 0.

  |ρ ε(x) -αρ l (x)| ≤ sup x∈H l, ε|ρ ε(x) -αρ l (x)|. Thus, it is enough to prove that (3.16) lim ε→0 sup x∈H l, ε|ρ ε(x) -αρ l (x)| = 0.

( 3 . 18 )

 318 |ρ ε (x ε ) -ρ ε (x)| ≤ |ρ sal ε (x ε ) -ρ sal ε (x)| + |ρ reg ε (x ε ) -ρ reg ε (x)| < 2C/5Thus, for any x ∈ I σ and x ε ∈ I σ , by (3.17),(3.18) and the assumption that|ρ ε (x ε ) -αρ l (x ε )| > C, we obtain |ρ ε (x) -αρ l (x)| ≥ |ρ ε (x ε ) -αρ l (x ε )| -|αρ l (x) -αρ l (x ε )| -|ρ ε (x ε ) -ρ ε (x)| ≥ C -2C/5 -2C/5 = C/5.(3.19)The estimate in(3.19) contradicts the fact that limε→0 ||ρ ε -ρ 0 || 1 = 0. Therefore,(3.16) holds and the lemma follows.

  Ωε µ ε(H l,ω )dθ ε(ω) = Ωε H l,ω ρ εdxdθ ε(ω) = Ωε H l,ω αρ l dxdθ ε(ω) + Ωε H l,ω (ρ ε -αρ l )dxdθ ε(ω) = α Ωε µ l (H l,ω )dθ ε(ω) + O sup ω∈Ωε sup x∈H l,ω |ρ ε(x) -αρ l (x)| Ωε m(H l,ω )dθ ε(ω).

( 3 . 21 )

 321 Ω ε µε(H l,ω )dθε(ω) = α Ω ε µ l (H l,ω )dθε(ω) + o(1) • Ω ε µ l (H l,ω )dθε(ω).

(3. 22 )

 22 Ω ε µε(Hr,ω)dθε(ω) = (1 -α) Ω ε µr(Hr,ω)dθε(ω) + o(1) • Ω ε µr(Hr,ω)dθε(ω).Using Lemma 3.7 together with equations (3.21) and(3.22) complete the proof of Theorem 3.4.

The existence of µ follows from the well known result of[START_REF] Lasota | On the existence of invariant measures for piecewise monotonic transformations[END_REF].

The usefulness of such a representation in studying stability and response theory of 1dimensional invariant densities was popularized by Baladi[START_REF] Baladi | On the susceptibility function of piecewise expanding interval maps[END_REF].

Later in the proof of (1) of Theorem 3.4, we will find α 1-α explicitly. In particular, we will show that l.a.h.r = α 1-α .

Acknowledgment We would like to thank anonymous referees for their suggestions which improved the presentation of the paper.

like to thank Jörg Schmeling for useful discussions on topics addressed in this work. W.B. would like to thank LMS for supporting a visit of S.V. to the Department of Mathematical Sciences at Loughborough University where this work was initiated. S.V. research was supported by the ANR "Perturbations", and by the CNRS-PEPS "Mathematical methods of climate models" and by the CNRS-PICS N. 05968. Part of this work was done while S.V. was visiting the Centro de Modelamiento Matemático, UMI2807, in Santiago de Chile with a CNRS support (délégation).


Carlo Baugé
email: carlo.bauge@ircam.fr

Mathieu Lagrange

Joakim Andén

Stéphane Mallat

REPRESENTING ENVIRONMENTAL SOUNDS USING THE SEPARABLE SCATTERING TRANSFORM

Keywords: scattering, separable scattering, wavelet, environmental sounds, similarity

Environmental sounds are an interesting subject of study for machine audition because of their wide variety of acoustical characteristics and their central presence in our everyday life. They are perceived effortlessly in the human auditory system whereas state-of-the-art computational systems are far from reaching the same efficiency.

In this paper we propose a novel representation of such sounds based on the scattering transform which has the property of stability to time-warping deformations and invariance to time-shift useful for classifications tasks. This representation is compared to several state-of-the-art approaches for the task of quantifying similarity between environmental sounds.

INTRODUCTION

Environmental sounds have been less studied than other types of sounds like music [START_REF] Tzanetakis | Musical genre classification of audio signals[END_REF] or speech [START_REF] Rabiner | Digital Processing of Speech Signals[END_REF]. Even though much is left to discover in the human auditory system, several modes of listening seem to be present, each one being presumably tuned to different types of input.

For example, in [START_REF] Gaver | What in the World Do We Hear?: An Ecological Approach to Auditory Event Perception[END_REF] Gaver distinguishes between two modes of listening used in different contexts and degrees of attention. The first one, called "everyday listening", corresponds to the way we usually hear in the day-to-day. Importance is given to source identification, localization in order to recognize which sources to interact with and which to avoid. This is opposed to the "musical listening" mode, which emphasizes the detailed characteristics of the sound itself like timbre or pitch.

The latter is therefore more analytical and precise than the former, which is rather used to compute a fast signature of what we hear. For environmental sounds, everyday listening seems the most appropriate type of listening. How can we design a representation that would be useful for implementing this mode of listening? What kind of acoustic features should be kept in the representation and which should be discarded?

To provide some answers to those questions, we propose a new computational approach based on the scattering transform [START_REF] Mallat | Group invariant scattering[END_REF] that has the property of time-shift invariance. In Section 2, we review the state-of-the-art in terms of sound representations and similarity computation. Section 3 presents the scattering transform and introduces a variant that adds frequency transposition property. Sections 4 and 5 describes the experimental protocol used to compare the representations and discusses the results.

RELATED WORK

The spectrogram, that is the modulus of the Short-Term Fourier Transform (STFT), is a commonly used representation of audio signals as it decomposes the signal into a time/frequency plane which has local time-shift invariance and loses little information when discarding the phase information. In order to better match this representation with the output of the cochlea and gain stability with respect to deformation, the frequency axis is often scaled logarithmically, for example using the mel scale.

In most audio processing applications, the features are fed to a classifier that operates better on low-dimensional and decorrelated features. It is therefore convenient to apply a final processing step to transform the logarithm of mel-scale spectra. Even though the Principal Component Analysis (PCA) could be used for this stage, similar decorrelation properties can be obtained (at least on speech and music data) by projecting the log-mel spectra on a cosine basis, which leads to the well-known Mel-Frequency Cepstral Coefficients (MFCC) [START_REF] Rabiner | Digital Processing of Speech Signals[END_REF][START_REF] Logan | Mel frequency cepstral coefficients for music modeling[END_REF].

Whatever the features chosen, their values evolve in time and this evolution carries information important to classifica- Several approaches exist in order to compare the representations of two sounds. In order to account for possible time-stretching between sounds, one can consider the Dynamic Time Warping (DTW) approach [START_REF] Rabiner | Digital Processing of Speech Signals[END_REF]. It aligns the two representations by finding an optimal match (with respect to cosine distance) between two given sequences (e.g. time series) with certain restrictions. The sequences are "warped" non-linearly in the time dimension to calculate a similarity measure independent of non-linear variations in the time dimension. Unfortunately, this non-linear matching has a complexity which is quadratic in the order of the number of frames in an audio sample.

A computationally less expensive approach is to model the diversity of the observations without preserving any sequentiality information. The observations are then considered as a Bag Of Frames (BOF) [START_REF] Aucouturier | The bagof-frame approach to audio pattern recognition: A sufficient model for urban soundscapes but not for poly-phonic music[END_REF] which is approximated using mixture models representing the probability distribution of the observations in the feature space. For that purpose, the Gaussian Mixture Model (GMM) is a convenient choice and has been considered for a wide variety of tasks [START_REF] Reynolds | Gaussian mixture models[END_REF].

PROPOSED METHOD

Ideally, one would have a computational paradigm where the timing relationship between events is explicitly modeled at the feature level which would lead to simple and efficient processing at the decision level, be it for similarity computation or for classification. The scattering transform1 [START_REF] Mallat | Group invariant scattering[END_REF], [START_REF] Andén | Multiscale scattering for audio classification[END_REF] yields such representation by cascading a wavelet filter bank and a modulus rectifier and has time-shift invariance property and stability to time-warping deformations.

Scattering operator

We consider a wavelet filter bank (ψ j ) j<J+P constructed by dilating a wavelet ψ whose octave bandwidth is 1/Q.

ψ j (t) = a -j ψ(a -j t) with a = 2 1/Q and j < J (1)
The filter bank consists of J logarithmically spaced filters where each ψ j covers the frequency interval [2Qπa -jπa -j , 2Qπa -j + πa -j ]. In the frequencies below 2Qπa -J , the filter bank consists of P (where P 1/ ln(a)) linearly spaced filters (for J ≤ j < J +P ) having the same frequency bandwidth as ψ J , which is 2πa -J .

To compute the first-order scattering coefficients, we apply the filter bank followed by a modulus rectifier to a signal x yielding a scalogram. We obtain the first-order coefficients by applying a low-pass filter Φ J to each frequency bin of the scalogram. This process is expressed in [START_REF] Rabiner | Digital Processing of Speech Signals[END_REF]. Low-pass filter Φ J accounts for frequencies below the lowest frequency of the filter bank. We obtain a representation similar to a melspectrum.

|x ψ j | Φ J (t), ∀j < J + P (2)
To recover lost information we compute what is called second-order coefficients. To do so, we consider each output |x ψ j1 (t)| for all j 1 < J + P as a new signal and apply the same process as for the first order:

||x ψ j1 | ψ j2 | Φ J (t), ∀j 1 , j 2 < J + P (3)
The second-order coefficients (3) give a frequential representation of the amplitude modulations over time for each frequency bin defined by the filter bank. For a signal of length N , there are Q log 2 (N/Q) first-order coefficients and Q 2 /2 log 2 2 (N/Q 2 ) second-order coefficients. Scattering has two relevant properties for us. The first is having the local time-shift invariance while keeping the amplitude modulations. Indeed, the information of the amplitude modulations inside a large time-window is lost in first-order coefficients but captured by second-order coefficients. The second property is the stability of the representation when the signal is slightly stretched in time as the logarithmic filter bank allows us to stabilize the high frequencies [START_REF] Mallat | Group invariant scattering[END_REF]. This approach share some of those properties with the spectrotemporal analysis proposed by Shamma &al [START_REF] Chi | Multiresolution spectrotemporal analysis of complex sounds[END_REF].

The scattering representation is the basis for the representations we considered for our experiments, which apply a transform on the scattering coefficients to add frequency transposition invariance.

Cosine Log Scattering (CLS)

Computing Cosine Log-Scattering [START_REF] Andén | Multiscale scattering for audio classification[END_REF] from scattering is similar to calculating MFCC from the mel spectrum. MFCCs are widely used on musical and voiced sounds, as they enable to easily separate the two components in a source-filter model: x(t) = e h(t). As mentioned earlier, the Discrete Cosine Transform (DCT) tends to concentrate the variance of the coefficients towards the low DCT frequencies.

Similarly to MFCCs, source and filter components can be linearly separated in the CLS representation. First-order CLS coefficients are computed by applying a DCT on the logarithm of first-order scattering coefficients. To compute second-order CLS coefficients at a time t 1 , we consider the logarithm of second-order scattering coefficients: log(||x

ψ j1 | ψ j2 | Φ J (t 1 )).
A DCT is applied first along j 2 and then along j 1 yielding a representation parametrised by k 1 and k 2 . These DCTs efficiently decorrelate the scattering coefficients [START_REF] Andén | Multiscale scattering for audio classification[END_REF].

By discarding high DCT-frequency coefficients, invariance to transposition and more stability to deformations is achieved. However, there is an intrinsic trade-off between loss of information and stability using this approach.

Separable scattering

To improve this trade-off, we propose to replace the log-DCT step by the application of the scattering transform anew on the scattering representation. Loss of information then does not come from discarding coefficients but from averaging them. We term this new form of scattering "separable" as it imply two separate steps of scattering. A first property is that this representation is invariant to frequency transposition within the support of the lowpass filter in the second scattering transform.

In addition, as the logarithmic scale in the scattering filter bank ensured stability to time deformations in the scattering representation, this same property is obtained for frequency deformations when the first-order scattering transform is applied anew to the scattering representation itself.

More precisely, a scattering transform is applied as would a log-DCT on first-order scattering coefficients. On secondorder coefficients the scattering transform is only applied along j 1 , in the direction of the dimension of the acoustic frequency which lead to a filtering operation similar to the one considered in [START_REF] Deviren | Frequency and wavelet filtering for robust speech recognition[END_REF].

EXPERIMENTS

Databases

We considered 3 databases [START_REF] Gygi | Similarity and categorization of environmental sounds[END_REF], [START_REF] Houix | A lexical analysis of environmental sound categories[END_REF] of environmental sounds, ranging from baby cries to airplane noises to footsteps to harp sounds. Within this database, the number of sounds per class was found to be too low compared to the diversity of the classes. We therefore decided to widen it to obtain a minimum of 4 sounds per class which lead to the "Gygi extended" database containing 294 sounds organised in the same 50 classes. The last database is called Houix (from [START_REF] Houix | A lexical analysis of environmental sound categories[END_REF]) and contains more specific sounds that occur in a kitchen. The 60 sounds are organised here into 4 classes (solid, gas, liquid and machine). Table 1 sums up the characteristics of the 3 databases.

Methods

For all the methods considered in this study, the amplitude of the sounds is first normalized using the Root Mean Square (RMS) in a 100-millisecond window around the peak amplitude of each sound.

The state-of-the-art representations used are the spectrogram, the mel-spectrum and the MFCCs. To compute the distances between sounds the Bag of Frames (BOF) algorithm is used [START_REF] Aucouturier | The bagof-frame approach to audio pattern recognition: A sufficient model for urban soundscapes but not for poly-phonic music[END_REF]. It learns a GMM model [START_REF] Reynolds | Gaussian mixture models[END_REF] for each sound and approximates the Kullback-Leibler divergence to compute a distance between sounds2 . A second distance algorithm used is the DTW which takes into account more temporal information on the sounds 3 .

For comparison, the scattering transform is applied on each sound with J such that Φ J will completely average the sound. This, in order to guarantee time-translation invariance of the representation. To obtain the same dimensionality for each sound, all sounds are zero-padded. This gives a vector length of 262144, 262144, and 524288 samples for the Gygi, GygiExt and Houix databases. In our case we used a quality factor Q of 8 in the scattering filter bank.

The CLS representation is computed by retaining only the first 10% lowest DCT-frequency first-order coefficients 10% of the second-order coefficients along k 1 and the 3 lowest DCT-frequency coefficients along k 2 . Indeed, performances are found to be maximal around this ratio. The separable scattering representation is obtained by applying the scattering transform with a quality factor of 1 and again averaging over the whole frequency-signal. This means that the representation is completely invariant to frequency transposition.

The global representation has the form of a vector containing both first-order and second-order coefficients. Three standard distance measures are considered: L 2 , L 1 norm, and cosine distance. Performance is evaluated by computing the Mean Average Precision over the resulting distance matrix [START_REF] Christopher | Introduction to Information Retrieval[END_REF].

RESULTS

Results are shown in Table 2. The first two columns show the results for a random distance matrix and the best results for the two state-of-the-art algorithms. To compute state-of-theart algorithms we varied sound representation (spectrogram, mel spectrum and MFCCs) and for BOF algorithm the number of Gaussians used per model varied from 1 to 20. Due to the short duration of the audio samples, the models that gave the best results were those with a low number of Gaussians, frim 1 to 3 depending on the database. The four last columns show the results for CLS and separable scattering when considering only first-order coefficients (CLS 1 and Sep 1 ) or both first-and second-order coefficients (CLS 1+2 and Sep 1+2 ).

If we compare the results of first-order with those of order 1 + 2, results are systematically better with order 1+2 for both CLS and separable representations. This indicates the importance of taking into account the amplitude modulations in the representation of sounds. As far as the kind of distance metric is concerned, the L 1 norm compares favorably to the others for CLS and the cosine distance for separable scattering. When considering first-and second-order separable scattering coefficients, cosine distance performs systematically better than the other distances.

The separable scattering also consistently shows better results than the CLS representation. This leads us to believe that the properties of invariance and stability of the separable scattering are useful in this task of similarity between environmental sounds.

More generally, the separable scattering representation shows better results than the two state-of-the-art algorithms with a stronger difference (+12.6% and +11%) for the two databases containing a wider variety of sounds (Gygi and GygiExt).

To summarize, scattering representations and in particular separable scattering seem to be more appropriate for our task than the usual representations such as mel-spectrum or MFCCs. It shall be noted that the distance measures we used for the two scattering representations were conceptually and algorithmically simpler than those of BOF and DTW.

CONCLUSION

We introduced a novel representation of environmental sounds by adding a property of invariance to frequency transposition in the scattering representation. Experimental results demonstrate its validity in the task of measuring similarity between environmental sounds on 3 databases. Future works will consider other types of sounds such as speech and music in order to better understand what kind of invariance is needed along the time and frequency axes with respect to the specificity of the data and the task at hand.

Fig. 1 .

 1 Fig. 1. Overview of the computation of CLS and separable scattering representations

Table 1 .

 1 consisting of short environmental sounds (up to 10 seconds) all sampled at 44.1 kHz. The first (called "Gygi") contains 100 sounds organised in 50 classes, each of 2 sounds. The sounds cover a wide variety Statistics of the 3 sound databases. These refer to the sound durations and the sound classes. Mean and standard deviation are shown. The last two columns show the total number of classes and elements.

		Durations(s)	Elements/cl		Total
		mean	σ	mean	σ	Cl. Elts.
	Gygi	2.32	0.86	2.00	0.00	50 100
	GygiExt	2.40	0.92	5.88	1.77	50 294
	Houix	2.92	1.98 15.00 13.83	4	60

Table 2 .

 2 Mean average precision results (in percentage) on each database for different algorithms. From left to right: random distances, state-of-the-art algorithms BOF and DTW, order 1 CLS and order 1+2 CLS, order 1 separable scattering and order 1+2 separable scattering.

		RAN D BOF DT W	L 1	CLS1 L 2	cos.	L 1	CLS1+2 L 2	cos.	L 1	Sep1 L 2	cos.	L 1	Sep1+2 L 2	cos.
	Gygi	5.1	31.8	25.8	26.8 23.3 21.6 37.2 31.9 33.1 19.9 20.8 30.0 35.8 35.4 44.4
	GygiExt	3.6	27.9	19.3	19.4 17.6 19.4 28.4 23.0 25.4 20.9 20.7 20.9 35.6 32.2 38.9
	Houix	43.6	54.6	55.5	55.0 54.8 55.0 53.2 52.0 52.6 57.4 57.3 52.0 56.1 56.8 59.0

MATLAB software for scattering and CLS can be found on http:// www.cmap.polytechnique.fr/scattering

MATLAB software for BOF can be found on http://www. jj-aucouturier.info/projects/mir/boflib.zip

MATLAB software for DTW can be found on http://labrosa. ee.columbia.edu/matlab/dtw/

This work has been partially supported by ANR-11-JS03-005-01and ANR grant BLAN 012601. Part of this work has been done while M. Lagrange was visiting IRCCYN (UMR 6597).