Romain Aimino
email: <aimino@cpt.univ-mrs.fr>.

Huyi Hu

Matt Nicol
email: <nicol@math.uh.edu>.

Andrei Török

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

Polynomial loss of memory for maps of the interval with a neutral fixed point

We give an example of a sequential dynamical system consisting of intermittent-type maps which exhibits loss of memory with a polynomial rate of decay. A uniform bound holds for the upper rate of memory loss. The maps may be chosen in any sequence, and the bound holds for all compositions.

Introduction

The notion of loss of memory for non-equilibrium dynamical systems was introduced in the 2009 paper by Ott, Stenlund and Young [START_REF] Ott | Memory loss for time-dependent dynamical systems[END_REF]; they wrote: Let ρ0 denote an initial probability density w.r.t. a reference measure m, and suppose its time evolution is given by ρt. One may ask if these probability distributions retain memories of their past. We will say a system loses its memory in the statistical sense if for two initial distributions ρ0 and ρ0, |ρt -ρt|dm → 0.

In [START_REF] Ott | Memory loss for time-dependent dynamical systems[END_REF] the rate of convergence of the two densities was proved to be exponential for certain sequential dynamical systems composed of one-dimensional piecewise expanding maps. Coupling was the technique used for the proof. The same technique was successively applied to time-dependent Sinai billiards with moving scatterers by Stenlund, Young, and Zhang [START_REF] Stenlund | Dispersing billiards with moving scatterers[END_REF] and it gave again an exponential rate. A different approach, using the Hilbert projective metric, allowed Gupta, Ott and Török [START_REF] Gupta | Memory loss for time-dependent piecewise expanding systems in higher dimension[END_REF] to obtain exponential loss of memory for time-dependent multidimensional piecewise expanding maps.

All the previous papers prove an exponential loss of memory in the strong sense, namely |ρ tρt |dm ≤ Ce -αt .

In the invertible setting, Stenlund [START_REF] Stenlund | Non-stationary compositions of Anosov diffeomorphisms[END_REF] proves loss of memory in the weak-sense for random composition of Anosov diffeomorphisms, namely

f • T n dµ 1 -f • T n dµ 2 ≤ Ce -αt
where f is a Hölder observable, T n denotes the composition of n maps and µ 1 and µ 2 are two probability measures absolutely continuous with respect to the Riemannian volume whose densities are Hölder. It is easy to see that loss of memory in the strong sense implies loss of memory in the weak sense, for densities in the corresponding function spaces and f ∈ L ∞ m . A natural question is: are there examples of time-dependent systems exhibiting loss of memory with a slower rate of decay, say polynomial, especially in the strong sense? We will construct such an example in this paper as a (modified) Pomeau-Manneville map:

T α (x) = x + 3 α 2 1+α x 1+α , 0 ≤ x ≤ 2/3 3x -2, 2/3 ≤ x ≤ 1 0 < α < 1. (0.1)
We use this version of the Pomeau-Manneville intermittent map because the derivative is increasing on [0, [START_REF] Aimino | Vitesse de mélange et théorèmes limites pour les systèmes dynamiques aléatoires et non-autonomes[END_REF], where it is defined, and this allows us to simplify the exposition. We believe the result remains true for time-dependent systems comprised of the usual Pomeau-Manneville maps, for instance the version studied in [START_REF] Liverani | A probabilistic approach to intermittency[END_REF]. * We will refer quite often to [START_REF] Liverani | A probabilistic approach to intermittency[END_REF] in this note. As in [START_REF] Liverani | A probabilistic approach to intermittency[END_REF], we will identify the unit interval [0, 1] with the circle S 1 , in such a way the map becomes continuous. We will see in a moment how an initial density evolves under composition with maps which are slight perturbations of (0.1). To this purpose we will define the perturbations of the usual Pomeau-Manneville map that we will consider.

The perturbation will be defined by considering maps T β (x) as above with 0 < β ≤ α. Note that T β = T α on 2/3 ≤ x ≤ 1. The reference measure will be Lebesgue (m). If 0 < β k ≤ α is chosen, we denote by P β k the Perron-Frobenius (PF) transfer operator associated to the map T β k .

Let us suppose φ, ψ are two observables in an appropriate (soon to be defined) functional space; then the basic quantity that we have to control is

|P βn • • • • • P β1 (φ) -P βn • • • • • P β1 (ψ)|dm. (0.2)
Our goal is to show that it decays polynomially fast and independently of the sequence P βn • • • • • P β1 : we stress that there is no probability vector to weight the β k . Note that, by the results of [START_REF] Sarig | Subexponential decay of correlations[END_REF], one cannot have in general a faster than polynomial decay, because that is the (sharp) rate when iterating a single map T β , 0 < β < 1.

In order to prove our result, Theorem 1.6, we will follow the strategy used in [START_REF] Liverani | A probabilistic approach to intermittency[END_REF] to get a polynomial upper bound (up to a logarithmic correction) for the correlation decay. We introduced there a perturbation of the transfer operator which was, above all, a technical tool to recover the loss of dilatation around the neutral fixed point by replacing the observable with its conditional expectation to a small ball around each point. It turns out that the same technique allows us to control the evolution of two densities under concatenation of maps if we can control the distortion of this sequence of maps. The control of distortion will be, by the way, the major difficulty of this paper.

Note that the convergence of the quantity (0.2) implies the decay of the non-stationary correlations, with respect to m:

ψ(x)φ • T βn • • • • • T β1 (x)dm -ψ(x)dm φ • T βn • • • • • T β1 (x)dm ≤ φ ∞ P βn • • • • • P β1 (ψ) -P βn • • • • • P β1 1 ψdm 1
provided φ is essentially bounded and 1( ψdm) remains in the functional space where the convergence of (0.2) takes place. In particular, this holds for C 1 observables, see Theorem 1.6. Conze and Raugy [START_REF] Conze | Limit theorems for sequential expanding dynamical systems on [0, 1[END_REF] call the decorrelation described above decorrelation for the sequential dynamical system T βn • • • • • T β1 . Estimates on the rate of decorrelation (and the function space in which decay occurs) are a key ingredient in the Conze-Raugy theory to establish central limit theorems for the sums

n-1 k=0 φ(T β k • • • • • T β1 x)
, after centering and normalisation. The question could be formulated in this way: does the ratio

n-1 k=0 [φ • T β k • • • • • T β1 (x) -φ • T β k • • • • • T β1 dm] n-1 k=0 φ • T β k • • • • • T β1 2
converge in distribution to the normal law N (0, 1)?

It would be interesting to establish such a limit theorem for the sequential dynamical system constructed with our intermittent map (0.1). Besides the central limit theorem, other interesting questions could be considered for our sequential dynamical systems, for instance the existence of concentration inequalities (see the recent work [START_REF] Aimino | Concentration inequalities for sequential dynamical systems of the unit interval[END_REF] in the framework of the Conze-Raugy theory), and the existence of stable laws, especially for perturbations of maps T α with α > 1/2, which is the range for which the unperturbed map exhibits stable laws [START_REF] Gouëzel | Central limit theorem and stable laws for intermittent maps Probab[END_REF].

We said above that we did not choose the sequence of maps T β according to some probability distribution. A random dynamical system has been considered in the recent paper [START_REF] Bahsoun | Decay of correlation for random intermittent maps[END_REF] for similar perturbations of the usual Pomeau-Manneville map. To establish a correspondence with our work, let us say that those authors perturbed the map T α by modifying again the slope, but taking this time finitely many values 0 < α 1 < α 2 < • • • < α r ≤ 1, with a finite discrete law. This random transformation has a unique stationary measure, and the authors consider annealed correlations on the space of Hölder functions. They prove in [START_REF] Bahsoun | Decay of correlation for random intermittent maps[END_REF] that such annealed correlations decay polynomially at a rate bounded above by n 1-1 α 1 . As a final remark, we would like to address the question of proving the loss of memory for intermittent-like maps, but with the sequence given by adding a varying constant to the original map, considered to act on the unit circle (additive noise). This problem seems much harder and a possible strategy would be to consider induction schemes, as it was done recently in [START_REF] Shen | On stochastic stability of expanding circle maps with neutral fixed points, Dynamical Systems[END_REF] to prove stochastic stability in the strong sense.

NOTATIONS. We will index the perturbed maps and transfer operators respectively as T β k and P β k with 0 < β k ≤ α. Since we will be interested in concatenations like P βn • P βn-1 • • • • • P βm we will use equivalently the following notations

P βn • P βn-1 • • • • • P βm = P n • P n-1 • • • • • P m .
We will see that very often the choice of β k will be not important in the construction of the concatenation; in this case we will adopt the useful notations, where the exponent of the P 's is the number of transfer operators in the concatenation:

P βn • P βn-1 • • • • • P βm := P n-m+1 m P n k = P k+n-1 • P k+n-2 • • • • • P k
In the same way, when we concatenate maps we will use the notation

T n-m+1 m := T n • T n-1 • • • • • T m instead of T βn • T βn-1 • • • • • T βm .
Finally, for any sequences of numbers {a n } and {b n }, we will write

a n ≈ b n if c 1 b n ≤ a n ≤ c 2 b n for some constants c 2 ≥ c 1 > 0.
The first derivative will be denoted as either T ′ or DT and the value of T on the point x as either T x or T (x).

1 The cone, the kernel, the decay Thanks to a general theory by Hu [START_REF] Hu | Decay of correlations for piecewise smooth maps with indifferent fixed points[END_REF], we know that the density f of the absolutely continuous invariant measure of T α in the neighborhood of 0 satisfies f (x) ≤ constant x -α , where the value of the constant has an expression in terms of the value of f in the pre-image of 0 different from 0. We will construct a cone which is preserved by the transfer operator of each T β , 0 < β ≤ α, and the density of each T β will be the only fixed point of a suitable subset of that cone.

We define the cone of functions

C 1 := {f ∈ C 0 (]0, 1]); f ≥ 0; f decreasing; X α+1 f increasing}
where X(x) = x is the identity function.

Lemma 1.1. The cone C 1 is invariant with respect to the operators

P β , 0 < β ≤ α < 1. Proof. Put T -1 β (x) = {y 1 , y 2 }, y 1 < y 2 ; put also χ β = 3 β y β 1 2 1+β
. Then a direct computation shows that

X α+1 P β f (x) = f (y 1 )y α+1 1 (1 + χ β ) α+1 1 + (1 + β)χ β + f (y 2 ) 3y 2 -2 y 2 α+1 y α+1 2 3 .
The result now follows since the maps

x → x α+1 f (x), x → χ β , x → y 1 , x → y 2
are increasing. The fact that α ≥ β implies the monotonicity of χ → (1+χ) α+1 1+(1+β)χ .

We now denote m(f ) = 1 0 f (x)dx and recall that for any 0 < β < 1 we have

m(P β f ) = m(f ). Lemma 1.2. Given 0 < α < 1, the cone C 2 := {f ∈ C 1 ∩ L 1 m ; f (x) ≤ ax -α m(f )}
is preserved by all the operators P β , 0 < β ≤ α, provided a is large enough.

Proof. Let us suppose that 1 0 f dx = 1; then we look for a constant a for which P β f (x) ≤ ax -α . Using the notations in the proof of the previous Lemma and remembering that

x α+1 f (x) ≤ f (1) ≤ 1 0 f dx = 1, we get P β f (x) = f (y 1 ) T ′ β (y 1 ) + f (y 2 ) T ′ β (y 2 ) ≤ ay -α 1 T ′ β (y 1 ) + y -α-1 2 T ′ β (y 2 ) = x y 1 α 1 T ′ β (y 1 ) + 1 a x α y α+1 2 T ′ β (y 2 ) ax -α , but x y 1 α 1 T ′ β (y 1 ) + 1 a x α y α+1 2 T ′ β (y 2 ) ≤ (1 + χ β ) α 1 + (1 + β)χ β + 1 a ( 3 2 y 1 ) α-β χ β (1 + χ β ) α ≤ (1 + χ β ) α 1 + (1 + β)χ β + 1 a ( 3 2 ) α χ β , ( * )
where the last step is justified by the fact that β ≤ α and 0 ≤ χ β ≤ 1/2. By taking the common denominator one gets

( * ) ≤ 1 + {β + [(α -β) + 2 α a -1 (β + 2)]}χ β 1 + (1 + β)χ β .
We get the desired result if (α

-β) + 2 α a -1 (β + 2) ≤ 1, which is satisfied whenever a ≥ 2 α (2 + α) 1 -α .
Remark 1.3. The preceding two lemmas imply the following properties which will be used later on.

1. ∀f ∈ C 2 , inf x∈[0,1] f (x) = f (1) ≥ min{a; [ α(1+α) a α ] 1 1-α }m(f ).

For any concatenation

P m 1 = P m • • • • • P 1 we have P m 1 1(x) ≥ min{a; [ α(1 + α) a α ] 1 1-α }.
See the proof of Lemma 2.4 in [START_REF] Liverani | A probabilistic approach to intermittency[END_REF] for the proof of the first item, the second follows immediately from the first. Remark 1.4. Using the previous Lemmas it is also possible to prove the existence of the density in C 2 for the unique a.c.i.m. by using the same argument as in Lemma 2.3 in [START_REF] Liverani | A probabilistic approach to intermittency[END_REF].

We now take f ∈ C 2 and define the averaging operator for ε > 0:

A ε f (x) := 1 2ε Bε(x) f dm
where B r (x) denotes the ball of radius r centered at the point x ∈ S 1 , and define a new perturbed transfer operator by

P ε,m := P nε m A ε = P βm+n ε-1 • • • • • P βm A ε
where n ε will be defined later on. It is very easy to see that

Lemma 1.5. For f ∈ C 2 P ε,m f -P nε m f 1 ≤ c f 1 ε 1-α
where c is independent of β.

Proof. By linearity and contraction of the operators P β we bound the left hand side of the quantity in the statement of the lemma by |A ε ff |dx and this quantity gives the prescribed bound as in Lemma 3.1 in [START_REF] Liverani | A probabilistic approach to intermittency[END_REF].

It is straightforward to get the following representation for the operator P ε,m :

P ε,m f (x) = 1 0 K ε,m (x, z)f (z)dz where K ε,m (x, z) := 1 2ε P nε m 1 Bε(z) (x).
We now observe that standard computations (see for instance Lemma 3.2 in [START_REF] Liverani | A probabilistic approach to intermittency[END_REF]), allows us to show that the preimages

a α n := T -n α,1 1 verify a α n ≈ 1 n 1 α ; here T -1 α,1
denotes the left pre-image of T -1 α , a notation which we will also use later on. Those points are the boundaries of a countable Markov partition and they will play a central role in the following computations; notice that the factors

c 1 , c 2 in the bounds c 1 1 n 1 α ≤ a α n ≤ c 2 1 n 1 α
depend on α (and therefore on β), but we will only use the a n associated to the exponent α; in particular we will denote by c α the constant c 2 associated to T α ; the dependence on α, although implicit, will not play any role in the following.

We will prove in the next section the following important fact.

• Property (P). There exists γ > 0 and n ǫ = O(ǫ -α ) such that for all ε > 0, x, z ∈ [0, 1] and for any sequence

β m , • • • , β m+nε-1 , one has K ε,m (x, z) ≥ γ.
We now show how the positivity of the kernel implies the main result of this paper.

Theorem 1.6. Suppose ψ, φ are in C 2 for some a with equal expectation φdm = ψdm. Then for any 0 < α < 1 and for any sequence

T β1 , • • • , T βn , n > 1, of maps of Pomeau-Manneville type (0.1) with 0 < β k ≤ α, k ∈ [1, n], we have |P βn • • • • • P β1 (φ) -P βn • • • • • P β1 (ψ)|dm ≤ C α ( φ 1 + ψ 1 )n -1 α +1 (log n) 1 α ,
where the constant C α depends only on the map T α , and

• 1 denotes the L 1 m norm.
A similar rate of decay holds for C 1 observables φ and ψ on S 1 ; in this case the rate of decay has an upper bound given by

C α F ( φ C 1 + ψ C 1 )n -1 α +1 (log n) 1 α
where the function F : R → R is affine.

Remark 1.7. One can ask what happens if we relax the assumption that all β n must lie in an interval [0, α] with 0 < α < 1. For instance, if the sequence β n satisfies β n < 1 and β n → 1, does the quantity P n 1 φ -P n 1 ψ 1 go to 0 for all φ, ψ in C 1 with φ = ψ? Similarly, what can we say when β n → 0? It follows from our main result that the decay rate of P n 1 φ -P n 1 ψ 1 is superpolynomial, but can we get more precise estimates for particular sequences β n , like β n = n -θ or β n = e -cn θ , θ > 0? We can also ask whether there is, in the case β n ∈ [0, α] covered by our result, an elementary proof for the decay to zero (without rate) of P n 1 φ -P n 1 ψ 1 . Proof of Theorem 1.6. We begin to prove the first part of the theorem for C 2 observables. We write n = kn ε + m with m < n ǫ . We add and subtract to the difference in the integral a term composed by the product of the first m usual PF operators and the product of k averaged operator P ε , each composed by n ε random PF operators; precisely we use the notation introduced above to get:

(LM ) := |P βn • • • • • P β1 (φ) -P βn • • • • • P β1 (ψ)|dm = |P n 1 (φ) -P ε,m+1+(k-1)nε • • • • • P ε,m+1 P m 1 (φ) + P ε,m+1+(k-1)nε • • • • • P ε,m+1 P m 1 (φ) -P ε,m+1+(k-1)nε • • • • • P ε,m+1 P m 1 (ψ) + P ε,m+1+(k-1)nε • • • • • P ε,m+1 P m 1 (ψ) -P n 1 (ψ)|dm. Thus (LM ) ≤ P n 1 (φ) -P ε,m+1+(k-1)nε • • • • • P ε,m+1 P m 1 (φ) 1 + P n 1 (ψ) -P ε,m+1+(k-1)nε • • • • • P ε,m+1 P m 1 (ψ) 1 + P ε,m+1+(k-1)nε • • • • • P ε,m+1 P m 1 (φ -ψ) 1 .
We now treat the first term I in φ on the right hand side ( the terms in ψ being equivalent), and we consider the last term III after that. We thus have:

I = P nε m+1+(k-1)nε • • • P nε m+1 P m 1 (φ) -P ε,m+1+(k-1)nε • • • • • P ε,m+1 P m 1 (φ) 1 .
To simplify the notations we put

       R 1 := P ε,m+1 , . . . R k := P ε,m+1+(k-1)nε , and
       Q 1 := P nε m+1 , . . . Q k := P nε m+1+(k-1
)nε , which reduce the above inequality to

I = (Q k • • • Q 1 -R k • • • R 1 )P m 1 (φ) 1 .
By induction we can easily see that

R k • • • R 1 -Q k • • • Q 1 = k j=1 k-j-1 l=0 R k-l (R j -Q j ) j-1 l=0 Q j-l-1
with R -1 = 1 and Q 0 = 1; by setting φ m := P m 1 (φ) and φm = P m 1 (φψ), we have therefore to bound by the quantity

k j=1 k-j-1 l=0 R k-l (R j -Q j ) j-1 l=0 Q j-l-1 φ m 1 . We now observe that Q j-l-1 φ m ∈ C 2 ; moreover R m g 1 ≤ g 1 ∀g ∈ C 2 , 1 ≤ m ≤ k, since R m
is a concatenation of transfer operators and the averaging map A ε which are all contractions on L 1 . Then we finally get, by invoking also Lemma 1.5,

I ≤ Q k • • • Q 1 φ m -R k • • • R 1 φ m 1 ≤ k j=1 c φ m 1 ε 1-α ≤ ck φ 1 ε 1-α .
We now look at the third term III which could be written as, by using the simplified notations introduced above:

III = R k • • • R 1 φm 1 .
By using Property (P) and by applying the same arguments as in the footnote 6 in [START_REF] Liverani | A probabilistic approach to intermittency[END_REF], one gets

R k • • • R 1 φm 1 ≤ e -γk φ -ψ 1 .
In conclusion we get

(LM ) ≤ ckε 1-α ( φ 1 + ψ 1 ) + e -γk ( φ 1 + ψ 1 ) ≤ c n n ε ε 1-α + e γ e -γ n nε ( φ 1 + ψ 1 ) ≤ C α ( φ 1 + ψ 1 )n 1-1 α (log n) 1 α having chosen ε = n -1 α log n ( 1 α -1)κ 1 α
, for a conveniently chosen κ.

In order to prove the second part of the theorem for C 1 observables, we invoke the same argument as at the end of the proof of Theorem 4.1 in [START_REF] Liverani | A probabilistic approach to intermittency[END_REF]. We notice in fact that if ψ ∈ C 1 then we can choose λ, ν ∈ R such that ψ λ,ν (x) = ψ + λx + ν ∈ C 2 , the dependence of the parameters with respect to the C 1 norm being affine.

For instance λ and ν could be chosen in such a way to verify the following constraints:

λ < -ψ ′ ∞ ; ν > max{ (1+α) ψ ∞ + ψ ′ ∞ -λ(2+α) 1+α , 1+a a-1 ψ ∞ - aλ 2(a-1) }.
2 Distortion: Proof of Property (P)

The main technical problem is now to check the positivity of the kernel; we will follow closely the strategy of the proof of Proposition 3.3 in [START_REF] Liverani | A probabilistic approach to intermittency[END_REF]. We recall that

2ε K ε,m (x, z) = P nε m 1 J (x)
where J = B ε (z) is an interval which we will take later on as a ball of radius ε around z.

By iterating we get (we denote with T -1 l,k , k = 1, 2, the two inverse branches of T l ):

2ε K ε,m = ln ε • • • l1 1 J (T -1 1,l1 • • • T -1 nε,ln ε x) |T ′ 1 (T -1 1,l1 • • • T -1 nε,ln ε x)T ′ 2 (T -1 2,l2 • • • T -1 nε,ln ε x) • • • T ′ nε (T -1 nε,ln ε x)| = ln ε • • • l1 1 J (x nε ) |T ′ 1 (x nε )T ′ 2 (T 1 x nε ) • • • T ′ nε (T nε-1 • • • T 1 x nε )|
where

x nε = T -1 1,l1 • • • T -1
nε,ln ε x ranges over all points in the preimage of x ∈ T nε • • • • • T 1 J. The quantity on the right hand side is bounded from below by

2ε K ε,m ≥ 1 Tn ε •••••T1(J) (x) inf z∈J 1 |T ′ 1 (z)T ′ 2 (T 1 z) • • • T ′ nε (T nε-1 • • • T 1 z)| .
We have therefore to control the ratio

inf z∈J 1 |T ′ 1 (z)T ′ 2 (T 1 z) • • • T ′ m (T m-1 • • • T 1 z)|
where m is the time needed for an interval J of length greater than 2ǫ to cover all the circle. We proceed as in the proof of Proposition 3.3 in [START_REF] Liverani | A probabilistic approach to intermittency[END_REF].

We need to introduce first some notations. Recall that a α n is the sequence of the preimages of 1 by the left branch of T α . We use similarly a β n for T β and define a 0 n as the infimum over all β > 0 of a β n . Remark that a 0 n is the sequence of the preimages of 1 by the left branch of the map T 0 defined by

T 0 (x) = 3x 2 , 0 ≤ x ≤ 2/3 3x -2, 2/3 ≤ x ≤ 1.
(2.3)

For k ≥ 1, we define the sequence a k n so that a k 0 = 1 and a k n is the preimage of 1 by T n k+1 the most at the left. In particular, a k n is the preimage of a k+1 n-1 by the left branch of T k+1 . Remark that a k n is a decreasing sequence in n and that a 0 n ≤ a k n ≤ a α n . We define the intervals

I k n = [a k n+1 , a k n ], which satisfy T n k+1 I k n = [ 2 3 , 1]. We also define I k n,+ = I k n+1 ∪ I k n = [a k n+2 , a k n ]
. We define the intermittent region I = [0, a 0 2 ] and the hyperbolic region H = [a 0 2 , 1].

Let J be an interval of length 2ǫ. We will iterate J under the non-stationary dynamics until it covers the whole space, and will control the distortion in the meantime.

At time k, the iterate K = T k 1 J verifies one of the following condition

1. K ∩ I = ∅;
2. K ∩ I = ∅, and K contains at most one a k ℓ , ℓ > 2;

3. K ∩ I = ∅, and K contains more than one a k ℓ , ℓ > 2.

Case 1. Suppose we are in the situation 1. Either one of the iterates of K will cross the point 2 3 where the maps are not differentiable, or it will fall in the situation 2 or 3. Let n ≥ 1 be the time spent before one of these situations occurs.

Since all maps are uniformly expanding on the hyperbolic region with uniformly bounded second derivatives, by standard computations, we have for all a, b ∈ K :

(T n k+1 ) ′ (a) (T n k+1 ) ′ (b) ≤ exp n-1 ℓ=0 sup ξ |T ′′ k+n-ℓ ξ| inf ξ |T ′ k+n-ℓ ξ| |T n+k-ℓ-1 k+1 a -T n+k-ℓ-1 k+1 b| .
Since 0 < β ≤ α < 1, the ratio

(T n k+1 ) ′ (a) (T n k+1 ) ′ (b) ≤ exp c 1 |T n k+1 (K)| ,
where c 1 = D 1-r depends only on α. After integration with respect to b, we find

|(T n k+1 ) ′ (a)| ≤ |T n k+1 (K)| |K| exp c 1 |T n k+1 (K)|
, from which we deduce

P n k+1 1 K ≥ 1 T n k+1 (K) |K| |T n k+1 (K)| exp -c 1 |T n k+1 (K)| .
If this new iterate of K intersects the intermittent region, we consider the situation 2 or 3, and continue the algorithm. If it is still in the hyperbolic region, but now contains the point 2 3 , we proceed in the following way. Let us call L = T n k+1 (K) the new iterate, and L l and L r the parts of the interval at the left and the right respectively of 2 3 . Either

|L l | > 1 3 |L| or |L r | > 1 3
|L|. In the first case, after one iteration, the image of L l will be contained in [ 2 3 , 1], with the right extremity at 1. So after say m steps, it will cover the whole unit interval, and the distortion is well controlled during this iteration. We then have

P m+1 k+n+1 1 L ≥ P m+1 k+n+1 1 L l ≥ 1 T m+1 k+n+1 (L l ) |L l | |T m+1 k+n+1 (L l )| exp -c 1 |T m+1 k+n+1 (L l )| ≥ 1 3 |L| exp {-c 1 } .
Setting n 1 = n + m + 1, we thus have

P n1 k+1 1 K = P m+1 k+n+1 P n k+1 1 K ≥ P m+1 k+n+1 1 L |K| |L| exp {-c 1 |L|} ≥ 1 3 |K| exp {-c 1 -c 1 |L|} .
In the second case, if the right part is longer than the left part, after one iteration, the iterate of the right part will be of the form [0, x], and we fall in the case 3 of the algorithm. We can apply the control on the distortion given in the case 3 to L r . Like in the previous case, doing this, we will get a factor 1 3 , but as we will see, the case 3 leads to the end of the algorithm, so we will meet the discontinuity point 2 3 at most one time during the whole procedure. Hence the factor 1 3 will appear only one time, and will not multiply itself several times, which could have spoiled the estimate.

Case 2. K is included in an interval I k ℓ,+ . Since T ℓ k+1 (I k ℓ,+ ) = [a k+ℓ 2 , 1] ⊂ [a 0 2 , 1
], after exactly ℓ iterations, the image of K will be included in the hyperbolic region, and we continue with the case 1. During this period of time, the distortion is controlled using the Koebe principle, that we recall below : Lemma 2.1 (Koebe Principle [5, Theorem IV.1.2]). For all τ > 0, there exists C = C(τ ) > 0 such that for all increasing diffeomorphism g of class C 3 with a non-positive Schwarzian derivative † , for all subintervals J 1 ⊂ J 2 such that g(J 2 ) contains a τ -scaled neighborhood of g(J 1 ) ‡ , one has

g ′ (x) g ′ (y) ≤ exp C |g(x) -g(y)| |g(J 1 )| for all x, y ∈ J 1 . † i.e. g ′′′ (x) g ′ (x) -3 2 g ′′ (x) g ′ (x) 2
≤ 0 ‡ i.e. the intervals on the left and on the right of g(J 1 ) in g(J 2 ) have length at least τ |g(J 1 )|

We apply it to g defined as the composition of the analytic extensions to (0, +∞) of the left branches of T k+ℓ , . . . , T k+1 with J 1 = I k ℓ,+ and J 2 = [δ, 2], where δ = δ(k, ℓ) is chosen small enough so that δ < a k ℓ+2 and T ℓ 0 δ < 1 2 a 0 2 . g has non-positive Schwarzian derivative since it is a composition of maps that have non-positive Schwarzian derivatives.

We have g(

J 1 ) = [a k+ℓ 2 , 1] ⊂ [a 0 2 , 1] and g(J 2 ) = [T ℓ k+1 δ, g(2)] ⊃ [T ℓ 0 δ, 2] ⊃ [ 1 2 a 0 2 , 2]. Set τ = min a 0 2 2(1-a 0 2 ) , 1 1-a 0 2
, which does not depends on the composition of maps, nor the number of steps ℓ.

The interval at the left of g(J 1 ) in g(J 2 ) contains [ 1 2 a 0 2 , a 0 2 ], and thus has length longer than 1 2 a 0 2 ≥ τ (1-a 0 2 ) ≥ τ |g(J 1 )|. Similarly, the interval at the right of g(J 1 ) in g(J 2 ) contains [START_REF] Aimino | Vitesse de mélange et théorèmes limites pour les systèmes dynamiques aléatoires et non-autonomes[END_REF][START_REF] Aimino | Concentration inequalities for sequential dynamical systems of the unit interval[END_REF] and thus has length longer than 1 ≥ τ (1a 0 2 ) ≥ τ |g(J 1 )|. We have proved that g(J 2 ) contains a τ -scaled neighborhood of g(J 1 ), so the Koebe principle implies there exists C = C(τ ) such that for all a, b ∈ J 1 one has

(T ℓ k+1 ) ′ (a) (T ℓ k+1 ) ′ (b) ≤ exp C |T ℓ k+1 (a) -T ℓ k+1 (b)| |T ℓ k+1 (J 1 )| ≤ exp c 2 |T ℓ k+1 (a) -T ℓ k+1 (b)| , with c 2 = 3C(τ ) since T k+ℓ k+1 (J 1 ) ⊃ [ 2 3 , 1]. As K ⊂ J 1 , one has for all a, b ∈ K (T ℓ k+1 ) ′ (a) (T ℓ k+1 ) ′ (b) ≤ exp c 2 |T ℓ k+1 (K)| ,
which implies [START_REF] Aimino | Vitesse de mélange et théorèmes limites pour les systèmes dynamiques aléatoires et non-autonomes[END_REF] and is of the form [a, 1], then we consider K ∩ [ 2 3 , 1] and case 1 will hold until we cover the whole interval, and we lose a factor 1 3 (only one time). Otherwise, we define ℓ -as the least integer such that I k ℓ-is included in K. We consider two sub-cases according to whether the part of K at the right of a k ℓ-is of length at least |K| 3 or not. In the first sub-case, we set

P ℓ k+1 1 K ≥ 1 T ℓ k+1 (K) |K| |T ℓ k+1 (K)| exp -c 2 |T ℓ k+1 (K)| . Case 3. If more than one third of K is in [ 2 3 ,
K ′ = K ∩ [a k ℓ-+1 , 1], which satisfies |K ′ | ≥ |K| 3 . Since K ′ ⊂ I k ℓ--1,+ and T ℓ--1 k+1 (K ′ ) ⊃ I k+ℓ--1 1
, we have by the step 2:

P ℓ--1 k+1 1 K ≥ 1 3 1 I k+ℓ --1 1 |K|e -c2 . Since T 2 k+ℓ-I k+ℓ--1 1
= [0, 1], and |T ′ β (x)| is bounded from above uniformly in β and x by some constant M > 0, we find

P ℓ-+1 k+1 1 K ≥ 1 3M 2 |K|e -c2 .
In the second sub-case, we choose K ′ in such a way that |K ′ | ≥ |K| 3 , the right extremity of K ′ is a k ℓ-and the left extremity is to the right of 0. We cut K ′ into pieces I k ℓ-, . . . , I k ℓ+ such that their union is of length longer than

|K ′ | 3 ≥ |K| 9
, with ℓ + minimal. This choice to cut K ′ rather than K allows us to estimate ℓ

+ : indeed, if we set K = [a, a k ℓ-], since ℓ + is minimal, the length of [a, a k ℓ+-1 ] is at least 2|K ′ | 3 ≥ 2|K| 9 . Hence, C (ℓ+-1) 1 α ≥ a k ℓ+-1 ≥ a k ℓ+-1 -a ≥ 2|K| 9
and consequently ℓ + = O(|K| -α ). By the computation done for the case 2, we have

P ℓ++1 k+1 1 K ≥ ℓ+ ℓ=ℓ- P ℓ++1 k+1 1 I k ℓ = ℓ+ ℓ=ℓ- P ℓ+-ℓ+1 k+ℓ+1 P ℓ k+1 1 I k ℓ ≥ ℓ+ ℓ=ℓ- P ℓ+-ℓ+1 k+ℓ+1 1 T ℓ k+1 (I k ℓ ) |I k ℓ | |T ℓ k+1 (I k ℓ )| exp -c 2 |T ℓ k+1 (I k ℓ )| . Since T ℓ k+1 (I k ℓ ) = [ 2 3 , 1
], which is sent after one iteration onto the whole interval, we have thanks to Remark 1.3 item 2

P ℓ++1 k+1 1 K ≥ ℓ+ ℓ=ℓ- P ℓ+-ℓ+1 k+ℓ+1 1 [ 2 3 ,1] |I k ℓ | 1/3 exp - c 2 3 ≥ c 3 3 exp - c 2 3 ℓ+ ℓ=ℓ- |I k ℓ | ≥ c 3 27 exp - c 2 3 |K|, with c 3 the constant given in Remark 1.3, since P ℓ+-ℓ+1 k+ℓ+1 1 [ 2 3 ,1] ≥ 1 3 P ℓ+-ℓ k+ℓ+2 1 ≥ c3 3 .
Conclusion. Let J be an interval of length at least 2ǫ. We associate to J a sequence of integers n 1 , m 1 , n 2 , m 2 , . . . , n p such that for n 1 steps the iterates of J is in the hyperbolic region (with possibly n 1 = 0), then for m 1 steps, it is in situation of the case 2, then it is again for n 2 steps in the hyperbolic region (recall that from the case 2, we can only fall into the case 1), and so on, until one iterate of J crosses the singularity 2 3 , or case 3 happens. These two situations lead to the end of the algorithm. We will only consider the situation where case 3 happens, and when the part of K to the right of a l l-has length not more than One has to estimate the supremum over all possible values of t = n 1 + m 1 + . . .+ n p + l + and shows it is of order ǫ -α . Let n ′ ǫ the minimal time needed for an interval of length at least 2ǫ to cover all the circle. We claim that n ′ ǫ = O(ǫ -α ), which concludes the proof since n 1 + m + . . . + n p ≤ n ′ ǫ , as after these iterations, J has not covered the circle, and l + = O(ǫ -α ), as we showed previously.

It remains to prove the claim. Since the first derivatives of all the T β is strictly increasing on the circle, the minimal time associated to intervals J of size 2ε, will be attained when an iterate of J will be located around 0, then moving according to case 3. We first consider an iterate whose length is one third of that of J (see above), located in (0, 2ε/3): we call this situation F. This implies a 1 d+t ≤ 2ε/3 which in turn shows the time needed to cover the circle is n ′′ ǫ = [ 3cα 2ε ] α . Take now J far from 0; if in n ′′ ε steps it will not meet the point 2/3, it will cover the circle, since the derivatives will be continuous along the path. Otherwise if it will meet 2/3 in a number of steps < n ′′ ε , the worst successive situation is to be sent in 0 in the situation F. In conclusion, the minimal time associated to intervals J of size 2ε will be bounded from above by 2n ′′ ε .

paper was completed during the Workshop Mathematics for Planet Earth. MN was supported by NSF grant DMS 1101315. AT was partially supported by the Simons Foundation grant 239583. The authors thank the referee for the useful comments. We are extremely grateful to Sébastien Gouëzel who pointed out a gap in the distortion argument of a preliminary version and suggested how to correct it by using the Koebe principle.

Note added in proof

A more careful analysis shows that in the proof of Property (P) the monotonicity of the derivatives is not necessary to estimate n ǫ . Thus, Theorem 2.6 holds for more general maps than (0.1), e.g. those in [START_REF] Liverani | A probabilistic approach to intermittency[END_REF]; the details can be found in [START_REF] Aimino | Vitesse de mélange et théorèmes limites pour les systèmes dynamiques aléatoires et non-autonomes[END_REF].

β

  x| and the quantity |T ′ β x| are bounded from above uniformly in β and x ∈ H respectively by D > 0 and 0 < r < 1. We then have

  (r n 2 +...+np +...+r n 2 )

|K| 3

 3 , the others being similar. For n ≥ n 1 + m 1 + . . . + n p + ℓ + + 1, we have

	P n 1 1 J ≥ P n1+...+ℓ++2 n-n1-...-ℓ+-1	P n1+m1+...+np+1 . . . P m1 ℓ++1 n1+1 P n1 1 1 J
	≥ P n1+...+ℓ++2 n-n1-...-ℓ+-1	P n1+m1+...+np+1 . . . P m1 ℓ++1 n1+1 1 T n 1 1 (J)

Acknowledgments

RA and SV were supported by the ANR-Project Perturbations and by the PICS ( Projet International de Coopération Scientifique), Propriétés statistiques des systèmes dynamiques déterministes et aléatoires, with the University of Houston, n. PICS05968. RA was supported by Conseil Général Provence-Alpes-Côte d'Azur. SV thanks the University of Houston for supporting his visits during the preparation of this work and the Newton Institute in Cambridge where this


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

AND J Örg Schmeling

Sandro Vaienti

ON TRANSFER OPERATORS AND MAPS WITH RANDOM HOLES

Keywords: Mathematics Subject Classification. Primary 37A05, 37E05 Transfer operators, equilibrium states, escape rates, Hausdorff dimension

We study Markov interval maps with random holes. The holes are not necessarily elements of the Markov partition. Under a suitable, and physically relevant, assumption on the noise, we show that the transfer operator associated with the random open system can be reduced to a transfer operator associated with the closed deterministic system. Exploiting this fact, we show that the random open system admits a unique (meaningful) absolutely continuous conditionally stationary measure. Moreover, we prove the existence of a unique probability equilibrium measure supported on the survival set, and we study its Hausdorff dimension.

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 orbit is terminated. Statistical aspects of such open systems have been addressed by many authors, see for instance [START_REF] Altmann | Leaking chaotic systems[END_REF][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]. Open dynamical systems have found interesting applications in physics [START_REF] Altmann | Leaking chaotic systems[END_REF][START_REF] Gaspard | Heat Conduction and Fouriers Law by Consecutive Local Mixing and Thermalization[END_REF][START_REF] Waalkens | Escape from planetary neighbourhoods[END_REF], and more recently, after the pioneering work 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], it was found that open systems are intimately related to studying metastable dynamical systems [START_REF] Bahsoun | Pseudo-orbits, stationary measures and metastability[END_REF][START_REF] Bahsoun | Metastability of certain intermittent maps[END_REF][START_REF] Bahsoun | Escape Rates Formulae and Metastability for Randomly Perturbed Maps[END_REF][START_REF] Dettmann | Open circle maps: Small hole asymptotics[END_REF][START_REF] Dolgopyat | The Diffusion Coefficient for Piecewise Expanding Maps of the Interval with Metastable States[END_REF][START_REF] Froyland | Metastability, Lyapunov exponents, escape rates, and topological entropy in random dynamical systems[END_REF][START_REF] González Tokman | Approximating invariant densities of metastable systems[END_REF] and their applications in geophysical sciences [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].

Following the work of [START_REF] Bahsoun | Escape Rates Formulae and Metastability for Randomly Perturbed Maps[END_REF], in this paper we study dynamical systems with random holes, called random open systems. One of the main motivation for studying random open systems is that they contribute to understanding the long-term statistics of random metastable systems. In [START_REF] Bahsoun | Escape Rates Formulae and Metastability for Randomly Perturbed Maps[END_REF] random perturbations of interval maps that initially admit exactly two invariant ergodic densities were studied. Under random perturbations, which generate random holes, leakage of mass between the two initially ergodic subsystems forces the random system to mix and admit a unique invariant density. It was shown in [START_REF] Bahsoun | Escape Rates Formulae and Metastability for Randomly Perturbed Maps[END_REF] that the invariant density of the random system can be approximated in the L 1 norm (with respect to Lebesgue measure), by a particular convex combination of the two invariant ergodic densities of the initial system. In particular, the weights in the convex combination is identified as the ratio of the escape rates from the left and right random open systems.

However, although almost every point (with respect to Lebesgue) from each initially ergodic subsystem escapes to the other ergodic component, some points survive in their initial set and do not visit the other ergodic component.

The main motivation of this paper is to study statistical properties of orbits which survive escaping from a random open system, and to determine the Hausdorff dimension of their set. From applications point of view, one can use our results to study statistical properties and dimension theory of orbits that do not visit other ergodic components in a random metastable system, such as the ones studied in [START_REF] Bahsoun | Escape Rates Formulae and Metastability for Randomly Perturbed Maps[END_REF], and to provide insight to geophysical models that study regions of the ocean with slow or poor mixing properties [START_REF] Dellnitz | Seasonal variability of the subpolar gyres in the Southern Ocean: a numerical investigation based on transfer operators[END_REF]. Mathematically, previous results on the Hausdorff dimension of the survival set were obtained in [START_REF] Glendinning | The doubling map with asymmetrical holes[END_REF][START_REF] Ferguson | Escape rates for Gibbs measures[END_REF][START_REF] Demers | Equilibrium states, pressure and escape for multimodal maps with holes[END_REF][START_REF] Liverani | Lasota-Yorke maps with holes: conditionally invariant probability measures and invariant probability measures on the survivor set[END_REF][START_REF] Sidorov | Supercritical holes for the doubling map[END_REF] for maps with deterministic holes. In our random setting, an interesting feature of our current work is that it uses tools from deterministic and closed systems to obtain results in random open systems: the transfer operator associated with our random open system can be reduced to a transfer operator associated with the closed deterministic system.

In section 2 we introduce the class of maps that we study, the associated transfer operator and the space of functions that it acts on, and known results from deterministic closed systems. In section 3 we introduce a class of maps with random holes, the notion of a conditionally stationary measure and prove a theorem highlighting its significance. Section 4 contains a topological characterization of the survival set of the random open system. In section 5 we introduce the transfer operator associated with the random system and show how to reduce it to a transfer operator of the original closed deterministic system. Exploiting this fact, we show that the random open system admits a unique (meaningful) absolutely continuous conditionally invariant measure. Moreover, we prove the existence of a unique probability equilibrium measure supported on the survival set, and we study its Hausdorff dimension. Section 6 contains two simple examples that highlight our results.

Setup and Preliminaries

2.1. Class of maps. Let I := [0, 1] and T : I → I be a Markov map; i.e., there exists a finite partition of open intervals P = {P i } l i=1 such that: (1) I = ∪ l i=1 P i , and

P i ∩ P j = ∅ when i = j; (2) ∃ 0 < α < 1 such that T |P i is C 1+α ; (3) if T (P i ) ∩ P j = ∅ then P j ⊂ T (P i ). (4) sup P ∈P (n) diam(P ) → 0 as n → ∞, where P (n) := ∨ n-1 i=0 T -i P.
We further assume that (5) ∃ N ∈ N such that, for all i = 1, . . . l, T N (P i ) = I.

Remark 2.1. Under assumptions (1)-( 4), it is well know that there exists a semi-conjugacy between a one-sided subshift of finite type (Σ, σ) and ([0, 1], T ). To avoid using (Σ, σ) and to keep our presentation mainly on [0, 1], we present some of the arguments using the map T (x) = 2x mod 1.

All the results of this paper are true for the class of maps introduced above in subsection 2.1.

2.2.

Space of functions and the transfer operator. We define our space of functions following the prescriptions at the beginning of Sect. 1 in [START_REF] Aaronson | Local Limit Theorems for Gibbs-Markov Maps[END_REF]. Let J := {T (P i )} l i=1 . Since J is contained in the sigma-algebra σ(P) generated by P, there exists a coarser partition G such that σ(J ) = σ(G). Let

H loc α := {φ : I → R| ∃C > 0, s.t. ∀x, y ∈ G j , G j ∈ G, |φ(x)-φ(y)| ≤ C|x-y| α }
For φ ∈ H loc α we let C α (φ) denote the semi-norm

C α (φ) = sup x =y∈G j G j ∈G |φ(x) -φ(y)| |x -y| α .
When equipped with the norm (ii) σ(L φ ) \ {λ} ⊂ B(0, r), with r < λ. (iii) There is a unique probability measure ν such that L * φ ν = ν; (iv) For all f ∈ H loc α we have

|| • || H loc α := C α (•) + || • || ∞ , H loc α is
lim n→∞ ||λ -n L n φ f -ρ I f dν|| ∞ = 0,
and the probability measure µ := gν is an equilibrium state associated with φ.

It is well known that the measure ν is also φ-conformal.

Perturbations and random holes

Let (ω k ) k∈N be an i.i.d. stochastic process with values in the interval S := [0, 1/2], and with probability distribution θ; we will set S := S ⊗N for the direct product of S upon which the direct product measure θ ⊗N is defined. We fix a point x 0 ∈ [0, 1] and consider "random" holes around it. More precisely, for ω ∈ [0, 1/2] we associate the interval I ω := (x 0ω, x 0 + ω). We call such an interval a random hole.

To explain the dynamics of the system with random holes, consider a finite path of the stochastic process, say, (ω 0 , ω 1 , . . . , ω k ). For this path, we first restrict T to I c ωo , then any point in I c ωo that gets mapped by T into I ω 1 , its orbit gets terminated. At time n = 2, any x ∈ I c ω 0 ∩ T -1 (I c ω 1 ) that gets mapped by T into I ω 2 its orbit gets terminated, and so on. The following measure, which was first introduced in [START_REF] Pianigiani | Expanding maps on sets which are almost invariant: decay and chaos[END_REF] in the determinstic setting and in [START_REF] Bahsoun | Escape Rates Formulae and Metastability for Randomly Perturbed Maps[END_REF] in the random setting, plays a central role in our analysis:

Definition 3.1. A Borel measure α on [0, 1] satisfying α(A) = 1 λ S dθ(ω) α(T -1 A ∩ T -1 I c ω ); where λ = S dθ(ω) α(T -1 I c ω ),
is called conditionally stationary measure. The escape rate, with respect to α, from the random open system is given byln λ. Moreover, we have

(3•1) λn α(A) = S dθ ∞ (ω) α(T -n A ∩ T -1 I c ω 1 ∩ T -2 I c ω 2 ∩ • • • ∩ T -n I c ωn ).
3.1. Significance of conditionally stationary measures. Consider the following random variable:

τ ω (x) = sup{i ≥ 0 : T i (x) ∈ I c ω i }. If τ ω (x) = n,
it means that, given the path ω, the orbit of x escapes through a random hole, I ω n+1 , exactly at time n + 1. We are interested in estimating the following expectation

(3•2) E θ [E m [τ ω (x)]].
Note that the first expectation is taken with respect to the ambient measure m. If the expectation in (3•2) is finite, it means that for almost every path ω, m-almost every point x will not survive.

The following theorem provides a useful estimate to (3•2). In particular it shows that whenever α is a conditionally invariant measure which is equivalent to m, for almost every path ω, m-almost every point x will not survive.

Theorem 3.2. If α is a conditionally stationary measure then:

(1) E θ [E α[τ ω (x)]] = λ 1- λ ; (2) E θ [E α[τ ω (x)] 2 ] -(E θ [E α[τ ω (x)]]) 2 = λ (1-λ) 2 . Proof. Given a path ω, set R n,ω := T -1 I c ω 1 ∩ T -2 I c ω 2 ∩ • • • ∩ T -n I c ωn . By (3•1), we have (3•3) E θ ( α(R n,ω )) = λn .
Therefore, by (3•3), we have

E θ [E α[τ ω (x)]] = E θ [ ∞ n=0 n( α(R n,ω ) -α(R n+1,ω ))] = ∞ n=0 n[E θ ( α(R n,ω )) -E θ ( α(R n+1,ω ))] = ∞ n=0 n( λn -λn+1 ) = λ 1 - λ . (3•4)
In the above computation we have used Tonelli's Theorem to exchange the sum and the expectation, and in the last step we have used the fact that

∞ n=0 n λn = λ (1 -λ) 2 .
Similarly, for (2), using the fact that

∞ n=0 n 2 λn = λ 1 + λ (1 -λ) 3 ,
we have

E θ [E α[τ ω (x)] 2 ] -(E θ [E α[τ ω (x)]]) 2 = ∞ n=0 n 2 (( λn -λn+1 ) - λ2 (1 -λ) 2 = (1 -λ) ∞ n=0 n 2 λn - λ2 (1 -λ) 2 = λ (1 -λ) 2 .
(3•5)

Remark 3.3. In [START_REF] Bahsoun | Escape Rates Formulae and Metastability for Randomly Perturbed Maps[END_REF], it was proved that piecewise expanding maps with random holes admit a conditionally invariant measure which is equivalent to Lebesgue. Moreover, the index of dispersion provided by Theorem 3.2

E θ [E α[τ ω (x)] 2 ] -(E θ [E α[τ ω (x)]]) 2 E θ [E α[τ ω (x)]
] plays a crucial role in approximating invariant densities of metastable random maps.

Conditionally invariant measures are quite tricky. In the case of deterministic holes, Demers and Young provided an example of a map with a hole that has infinite number of absolutely continuous conditionally invariant measures (accim) with overlapping supports [START_REF] Demers | Escape rates and conditionally invariant measures[END_REF]. Moreover, all the accims in the example of [START_REF] Demers | Escape rates and conditionally invariant measures[END_REF] have the same escape rate. However, it is suggested in [START_REF] Demers | Escape rates and conditionally invariant measures[END_REF] that only 'natural' accims are meaningful. For instance, 'natural' maybe in the sense that the density of the accim belongs to a certain class of functions that include the constant density, such that under the iterates of the conditional transfer operator each function in this class converges in the appropriate topology to the density of the 'natural' accim (see section 5.2 of [START_REF] Demers | Escape rates and conditionally invariant measures[END_REF]).

In [START_REF] Bahsoun | Escape Rates Formulae and Metastability for Randomly Perturbed Maps[END_REF] we studied maps with random holes in the case where the potential of the transfer operator isln |T ′ (x)| ∈ BV , where BV is the space of functions of bounded variation on the unit interval. For sufficiently small holes, using the perturbation result of [START_REF] Keller | Stability of the spectrum for transfer operators[END_REF], we proved in [START_REF] Bahsoun | Escape Rates Formulae and Metastability for Randomly Perturbed Maps[END_REF] the existence of a unique absolutely continuous conditionally stationary measure whose density belongs to BV , and which is natural along the lines above. In the current work, the transfer operator has a general Hölder potential, and there is no smallness condition on the size of the holes. In Theorem 5.5 and Corollary 5.6 we obtain a unique absolutely continuous conditionally stationary measure that satisfy the criteria of section 5.2 of [START_REF] Demers | Escape rates and conditionally invariant measures[END_REF].

Topological description of the surviving set

We now introduce the random function

1(x, ω) = 1 x ∈ I ω 0 else . Clearly P θ (1(x, ω) = 1) = ∞ -∞ 1(x, ω) dθ(ω) = θ(|x -x 0 | ≥ ω) = |x-x 0 | 0 dθ = F θ (|x -x 0 |), (4•6)
where F θ is the distribution function of ω (We note that here F θ (0) = θ({0}).

Let ω = ω 0 ω 1 • • • ω n • • • ∈ [0, 1] N .
We introduce the i.i.d. (for fixed x) stochastic functional process

g n (x, ω) := 1(x, ω n ).
The function g n (T n x, ω) describes whether the trajectory of a point x falls into a hole at time n or not.

We are interested in the set of points that avoid, under the dynamics of T , the random holes, i.e. those trajectories that "survive". We first start with a topological description.

Given a realization ω ∈ [0, 1] N we define its surviving set as

S ω := {x ∈ [0, 1] : g n (T n x, ω) = 1 ∀n ∈ N} .
The potential surviving set is defined as

S := [0, 1] \ {x ∈ [0, 1] : x ∈ S ω for θ ⊗N -a.e. ω}. Theorem 4.1. Let supp θ = [a, b]. Then S = m∈N T -m n∈N T -n ([0, 1] \ (x 0 -b, x 0 + b)) .
Proof. If a point's orbit visits the interval (x 0b, x 0 + b) only finitely many times its surviving probability is positive. Hence the inclusion "⊇" is clear. Fix ε > 0 and x ∈ [0, 1]. If the trajectory of x hits the interval (x 0b + ε, x 0 + bε) infinitely often the Borel Cantelli Lemma implies that it will eventually fall into a hole since P θ (g n (T n x, ω) = 0) is, uniformly in n, bounded away from zero as long as |T n x-x 0 | < b-ε. This proves "⊆". Remark 4.2. For T (x) = 2x mod 1, it is easy to check that if b < 1/8, then the potential surviving set is non-empty. In fact if b < 1/8 the hole (x 0b, x 0 + b) is contained in two adjacent binary intervals of length 1/4. In the binary coding these two intervals correspond (for each interval separately) to fixing the first two digits, i.e. a cylinder of length 2. Now the set of symbolic sequences x ∈ {0, 1} N that do not contain any two a priori fixed words out of [00], [01], [START_REF] Demers | Equilibrium states, pressure and escape for multimodal maps with holes[END_REF], [START_REF] Demers | Escape rates and conditionally invariant measures[END_REF] as a sub-word contains the sequences 0 ∞ , 1 ∞ or (01) ∞ .

Statistical aspects of the surviving trajectories

To study the long-term statistics of the surviving trajectories, we first introduce the averaged transfer operator associated with the system with random holes. We will introduce in this section a interesting condition1 that allows us to reduce the transfer operator associated with the system with random holes to a transfer operator (with a new potential) associated with the deterministic and closed system T . 5.1. The averaged transfer operator. In random dynamical systems [START_REF] Kifer | Ergodic theory of random transformations[END_REF], it is often useful to study the average transfer operator of the random system, which is the integral, with respect to the noise, of the transfer operators associated with the perturbed maps. Here in our setting, the map is fixed at all times. However, a hole is randomly selected at each time step.

Let φ ∈ H loc α . The averaged transfer operator with potential φ, associated with the system with random holes and acting on H loc α , is defined by:

L φ (f )(x) := S L φ (f • 1(•, ω))(x) dθ(ω).
Its dual operator, which acts on the space of finite measures, will be denoted by L φ *

. Notice that without further assumptions on the hole I ω the operator L φ (f ) will not leave the space H loc α invariant. We provide in this way; we first recall the definition of P θ in (4•6) and we write g(x) := P θ (1(x, ω) = 1) and ψ(x) := log g(x).

Then we will make the following Assumption: ψ ∈ H α .

When does ψ ∈ H

(x) -g(y)| = |F θ (|x -x 0 |) -F θ (|y -x 0 |) = |x-x 0 | 0 dθ - |y-x 0 | 0 dθ = |x-x 0 | |y-x 0 | dθ = θ(I x,y ) ≤ K(diam I x,y ) α ≤ K|x -y| α . If x = x 0 then |g(x 0 ) -g(y)| = θ({0}) - |y-x 0 | 0 dθ = θ({0}) -θ(0) + lim ε→0 + |y-x 0 | ε dθ = lim ε→0 + θ((ε, |y -x|) ≤ K|x -y| α .
If in addition θ({0}) > 0 we have that F θ (|xx 0 |) > 0 for all x ∈ [0, 1] and hence g > 0. That together with the Hölder continuity of g implies that ψ = log g is Hölder continuous. Remark 5.4. θ({0}) > 0 has a physical interpretation. It means that the system randomly switches on a random hole and it has the possibility of not switching on any hole at all. More precisely, at any moment n the system is closed (no trajectory falls into a hole) with positive probability.

Conditionally invariant measures and equilibrium states.

Theorem 5.5. Assume θ is Ahlfohrs upper semi-regular. For ψ ∈ H loc α we have

(i) For f ∈ H loc α we have L φ (f ) = L φ (f • g) and L φ (f ) = L φ+ψ (f ). (ii) supp g = [0, 1] \ (x 0 -a, x 0 + a) where supp θ = [a, b].
(iii) L φ : H loc α → H loc α has a dominant simple eigenvalue λ and its corresponding eigenfunction, ρ, is strictly positive. There is a unique eigenmeasure ν with L φ * (ν) = λν. The associated measure is given by μφ,θ := ρν. (iv) supp μφ,θ = n∈N T -n ([0, 1] \ (x 0a, x 0 + a)) (v) The associated equilibrium state to the potential φ + ψ is fully supported.

Proof.

(i) By definition we have

L φ (f ) = S T y=x e φ(y) f (y)1(y, ω) dθ(ω) = T y=x e φ(y) f (y)P θ (1(y, ω) = 1) = L φ (f • g) = L φ+ψ (f ). (ii) We have 0 = g(x) = F θ (|x -x 0 |) ⇐⇒ |x -x 0 | < a (iii) Follows from (i) and Proposition 2.2. (iv) We notice that with S n φ(z) = n-1 k=0 φ(T k z) L φ n (f ) = T n y=x e Snφ(y) n-1 k=0 g(T k y)f (y) .
Hence, for all n ∈ N

supp ρ = supp L φ (ρ) = supp L φ n (ρ) = n-1 k=0 supp g • T k . But ∞ k=0 supp g • T k = n∈N T -n ([0, 1] \ (x 0 -a, x 0 + a)) .
(v) This follows from the previous arguments and the fact that the Hölder continuity of ψ = log g implies that supp g = [0, 1].

Using (iii) of Theorem 5.5, we have Corollary 5.6. The measure α given by

α(A) := 1 λ X S 1 A 1 I c ω ρdθdm
is a conditionally stationary measure which is equivalent to m. Its associated escape rate is given byln λ, where

λ = S dθ(ω) X ρ1 I c ω dm.
Moreover, for all f ∈ H loc α we have

lim n→∞ || λ-n L φ n f -ρ I f dν|| ∞ = 0.
Proof. We L φ ρ = λρ, with I ρdm = 1. By using the definition of L φ , it follows that: λ = S dθ(ω)

I ρ1 I c ω dm.
By using again the fact that L φ ρ = λρ, with I ρdm = 1 and the definition of L φ , we see that the above defined measure α satisfies the following:

α(A) = 1 λ S dθ(ω) α(T -1 A ∩ T -1 I c ω ); λ = S dθ(ω) α(T -1 I c ω ); and λn α(A) = S dθ ∞ (ω) α(T -n A ∩ T -1 I c ω 1 ∩ T -2 I c ω 2 ∩ • • • ∩ T -n I c ωn ).
The fact that for all f ∈ H loc

α lim n→∞ || λ-n L φ n f -ρ I f dν|| ∞ = 0
follows from Theorem 5.5 and (iv) of Proposition 3.2.

Remark 5.7. Since α is equivalent to m, by (i) of Theorem 3.2, for almost every ω, m almost every x will not survive. This yields to the natural question: what is the 'dimension' of the set of points that never escape through the random holes?

5.4. Properties of μφ,θ . The relation between L φ and L φ+ψ yields that the associated measure μφ,θ is the usual equilibrium state µ φ+ψ , for the closed deterministic system T , with respect to the Hölder continuous potential φ + ψ. It is well known (see for instance [START_REF] Baladi | Positive transfer operators and decay of correlations[END_REF]) that the measure µ φ+ψ has the Gibbs property

C -1 < µ φ+ψ ([x 1 • • • x n ]) e Snφ(x)+Snψ(x) [y 1 •••yn] e Snφ(y)+Snψ(y) < C
where C > 1 and [z 1 • • • z n ] denotes the set of all numbers having z 1 • • • z n as their first dyadic digits. We also call μφ,θ the measure of conditional surviving probability 2 .

2 Let us elaborate on calling μφ,θ the measure of conditional surviving probability.

Given a point y ∈ [0, 1] and a finite path of its trajectory y 1 , y 2 , • • • , y n . Then e Snφ(y) is up to normalization the probability with respect to the equilibrium state µ φ of choosing this particular path. On the other hand e Snψ(y) is up to normalization the probability that y survives along this particular path. Since choosing the path is independent of 5.5. Asymptotic behaviour and the Hausdoff dimension. When the set 3 S = ∅, we consider three families of potentials: t(φ + ψ), φ + tψ and tφ + T (t)ψ where in the latter case T (t) is defined by sup ν-ergodic {h ν + [0,1] (tφ + T (t)ψ) dν} = 0. We are interested in the asymptotics of the corresponding equilibrium states as t → ∞.

Theorem 5.8. Any accumulation measure µ S ∞ of µ t(φ+ψ) as t → ∞ has support on the set of trajectories with the largest possible Birkhoff average of φ+ψ, i.e. for µ S ∞ -a.e. x the Birkhoff average lim n→∞ n-1 k=0 φ(T k x)+ψ(T k x) is maximal. Moreover it has maximal entropy on this set.

Proof. The statement of the theorem follows from standard results in ergodic optimization [START_REF] Jenkinson | Ergodic optimization[END_REF]. Any accumulation point maximizes the integral [0,1] (φ + ψ) dν and is hence supported on the set with maximal possible ergodic averages. Moreover, any such accumulation point has maximal entropy of all invariant measures supported on this set.

Remark 5.9. The maximal Birkhoff average of φ + ψ can be interpreted in the following way: the maximal averages are obtained on the set whose generic µ φ trajectory will have the highest joint probability of being chosen and of survival. Then this maximizing measure µ S ∞ is concentrated on the set of trajectories where the probability of being chosen and of survivinal is balanced in the highest way.

Theorem 5.10. Any accumulation measure µ S ∞ of µ (φ+tψ) as t → ∞ has support on S. Moreover,

sup{h ν + [0,1] φ dν : ν-invariant and ν(S) = 1} = h µ S ∞ + [0,1] φ dµ S ∞ .
Therefore it is an equilibrium state with respect to the potential φ on the surviving set S.

Proof. By the variational principle sup

ν-invariant h ν + [0,1] φ dν + t [0,1] ψ dν = h µ φ+tψ + [0,1] φ dµ φ+tψ + t [0,1]
ψ dµ φ+tψ .

activating the holes we get μφ,θ ([

y 1 • • • y n ]) ∼ µ φ ([y 1 • • • y n ]) • P θ (y survives) [y1•••yn] µ φ ([y 1 • • • y n ]) • P θ (y survives) ∼ µ φ ([y 1 • • • y n ]) • P θ (y survives) E µ φ (y survives up till time n) .
The expectation in the denominator decays exponentially in n. Hence the measure μφ,θ ([y 1 • • • y n ]) is the conditional probability of a cylinder to be chosen and surviving rescaled with the average probability of a cylinder of length n to survive under the condition that all trajectories survive until time n.

3

Recall that in Remark 4.2 it is verified that for T (x) = 2x the set S = ∅.

Since φ < ∞ and h ν ≤ log 2 it follows that µ S ∞ maximizes (over all invariant measures) the integral [0,1] ψ dν. Let

S := n∈N T -n ([0, 1] \ (x 0 -b, x 0 + b)).
This set is the set of points that never (not only in finite time) enter the critical region and hence have surviving probability 1. We also remark that any invariant measure on S is indeed concentrated on S. Since S = ∅ and S is compact and forward invariant there is a maximizing measure ν supported on S with [0,1] ψ dν = 0. Let B ∩ S = ∅. Then µ S ∞ (B) = 0. Otherwise since g| B < 1 the integral of ψ would be negative. Hence µ S ∞ ( S) = 1 and consequently µ S ∞ (S) = 1. Assume that there is an invariant (and hence also an ergodic) measure ν supported on S and an ε > 0 with

(5•7) h ν + [0,1] φ dν > h µ S ∞ + [0,1] φ dµ S ∞ + ε.
Since [0,1] ψdν = 0, we have

(5•8) h ν + [0,1] φ dν + t [0,1] ψ dν = h ν + [0,1]
φ dν.

Using (5•7) and (5•8), we get

(5•9) h ν + [0,1] φ dν + t [0,1] ψ dν > h µ S ∞ + [0,1] φ dµ S ∞ + ε.
Then for sufficiently large t, and by using the fact that t [0,1] ψ dµ φ+tψ ≤ 0, we have

h ν + [0,1] φ dν + t [0,1] ψ dν > h µ φ+tψ + [0,1] φ dµ φ+tψ + ε 2 ≥ h µ φ+tψ + [0,1] φ dµ φ+tψ + t [0,1] ψ dµ φ+tψ + ε 2 .
This contradicts the variational principle.

For the interpretation of the last family we consider the symbolic representation Σ = {0, 1} N corresponding to the dyadic expansions of the real numbers in the interval. We introduce a metric on Σ by

diam([x 1 • • • x n ]) := S n ψ(x)
where ψ := log g [0,1] g dx . Then this space Σ becomes a Cantor set. The pointwise dimension

d φ,θ (x) = log µ φ ([x 1 • • • x n ]) diam([x 1 • • • x n ]
) at a point x is a measure of the deviation from the expected dying out probability of a given path.

From standard multifractal analysis [START_REF] Pesin | Dimension theory in dynamical systems[END_REF] we know Theorem 5.11.

dim H {x : d φ,θ (x) = -T ′ (t)} = T (t) + T ′ (t) • t = dim H µ tφ+T (t)ψ .

Examples

In the previous sections, we studied the dimension of the survival set under the condition that θ({0}) > 0. To interpret what this means, one can imagine that the random holes are possible 'gates' where orbits under the dynamics of the map T escape. The condition θ({0}) > 0 means that in certain situations all gates are closed, and thus no orbit escape. In fact this interpretation explains that the condition θ({0}) > 0 provides interesting dynamics that cannot happen in a deterministic setting such as in the systems studied in [START_REF] Liverani | Lasota-Yorke maps with holes: conditionally invariant probability measures and invariant probability measures on the survivor set[END_REF]. Moreover, we believe that the condition θ({0}) > 0 may provide interesting examples of survival sets that are 'fat'. Having said that, it would be also natural to ask what happens when θ({0}) = 0? We present some examples to shed some light on this direction. Since in the examples below, the measure θ is discrete, unlike, in Theorem 5.5, we first state an analogous result to that of Theorem 5.5 in a simple discrete case.

• There are K holes, K ≥ 2, denoted by {H ω i } K i=1 and selected independently according to {p i } K i=1 ; i.e., p i > 0 and K i=1 p i = 1;

• {H ω i } K-1
i=1 are elements of the Markov partition of T ; • H K = ∅ (the case of no hole4 ); • Hloc α := {Locally Hölder functions w.r.t. the Markov partition of T }.

The following result leeds to an analogous conclusion as that of Theorem 5.5. Proposition 6.1. Let φ ∈ Hloc α . Then for any f ∈ Hloc

α we have L φ (f ) = L φ+ψ (f ), with ψ ∈ Hloc α . Proof. Let X ω i = [0, 1] \ H ω i . We have L φ (f )(x) = K i=1 p i L φ (1 Xω i f )(x) = L φ ( K i=1 p i 1 Xω i • f )(x) = L φ+ψ (f ), where ψ(x) = log K i=1 p i 1 Xω i (x) . Note ψ ∈ Hloc α because K i=1 p i 1 Xω i ∈ Hloc α and K i=1 p i 1 Xω i (x) > 0 since 5 H K = ∅. and each a i ∈ {1, 2} ≡ {[0, 1/2), [1/2, 1)}. Notice that ( p 1 + p ) n ≤ μφ,θ (Z(j, a 0 , a 1 , . . . , a n-1 )) ≤ ( 1 1 + p ) n .
Thus, we conclude that the support of μφ,θ is a fat set.

Example 6.3. We now consider another example to get a sense on the difference between the case 'θ({0}) > 0' (as in Example 6.2) and the case when 'θ({0}) = 0': T (x) = 3x mod 1. Let H 1 = (0, 1/3) and H 2 = (0, 2/3) be two holes. We now consider random holes in the following way: at each time n, we either have a 'small' hole To find the measure μφ,θ , we consider  * φ , the transpose of the matrix defined in (6•12). Let Q be the stochastic matrix defined by

Q ij = ( Â * φ ) ij ρj λρ i
whose stationary probability vector q is defined by Then μφ,θ , defined on cylinder sets, is given by μφ,θ (Z(j, a 0 , a 1 , . . . , a n-1 )) = q a 0 • Q a 0 a 1 • • • Q a n-2 a n-1 with μφ,θ (Z(j, a 0 )) = q a 0 , and each a i ∈ {1, 2, 3} ≡ {[0, 1/3), [1/3, 2/3), [2/3, 1)}. Notice that 0 ≤ μφ,θ (Z(j, a 0 , a 1 , . . . , a n-1 )) ≤ ( 1 1 + p

q i = ρi • νi ;
) n .

In particular, for any Z(j, a 0 , a 1 , . . . , a n-1 ) such that a i = 1 for some j ≤ i ≤ j + n -1 we have μφ,θ (Z(j, a 0 , a 1 , . . . , a n-1 )) = 0. Thus, we conclude that the support μφ,θ is a thin set. S. Vaienti: Aix Marseille Université, CNRS, CPT, UMR 7332, 13288 Marseille, France and Université de Toulon, CNRS, CPT, UMR 7332, 83957 La Garde, France.

E-mail address: vaienti@univ-mrs.fr

Remark 5 . 2 .Theorem 5 . 3 .

 5253 α ? And what does it physically mean? Definition 5.1. A measure θ on [0, +∞) is called Ahlfohrs upper semiregular if there is a constant K > 0 and a real number α > 0 such that for all non-empty open intervals I ⊂ [0, +∞) θ(I) (diam I) α < K. Ahlfors upper semi-regular measures include fractal measures like the measures of maximal dimension of dynamically defined Cantor sets, i.e. Cantor sets that arise from smooth expanding repellers. The function g is Hölder continuous if θ is Ahlfors upper semi-regular. If in addition θ({0}) > 0 then ψ is Hölder continuous. Proof. Let x, y = x 0 and I x,y = (|xx 0 |, |yx 0 |) ⊂ R + . Then diam I x,y ≤ |x -y|. If θ is Ahlfors upper semi-regular we conclude |g

3 .I Ω 1 A 1

 311 H 1 with probability p, or a bigger hole H 2 with probability 1p. Again the transfer operator Lφ , with φ(x) =ln |T ′ x|, can be represented by the following matrix (acting by multiplication from the left):The corresponding left and right eigenvectors are, respectively, Note that for any measurable set A, the absolutely continuous conditionally invariant measure α is given byα(A) = 1 λ Xω ρdθ(ω)dm = μφ,θ (A) = 3p p + 1 (m(A ∩ [1/3, 2/3])) + 3 p + 1 (m(A ∩ [2/3, 1])) .

  (p + 1) 1/(p + 1) 0 p/(p + 1) 1/(p + 1) 0 p/(p + 1) 1/(p + 1)   , q = [0 p/(p + 1) 1/(p + 1)].

Remark 6 . 4 .

 64 The above examples suggest the following interesting problem: In Example 6.2, where θ({0}) > 0 the measure μφ,θ is supported on a fat subset of the interval [0, 1]. In Example 6.3, where θ({0}) = 0, the measure is only supported on a thin subset of [1/3, 1]. It would be interesting to work out analogous theorems to those in the previous sections in the case when θ(0) = 0. Based on the above examples, we believe that for the same map T , one may get a 'fat' survival set when θ({0}) > 0; however, when θ({0}) = 0 the survival set would be 'thinner'.

  the Banach space of locally Hölder continuous functions with exponent α. If a function f : I → R is Hölder with exponent α, on the whole interval I, we simply write f ∈ H α .

	2.3. Conformal measure. Let A be a Borel measurable set. A measure
	m is called φ-conformal if
	m(T (A)) =	e -φ dm
	A
	whenever T : A → T (A) is injective.
	2.4. Known results about the deterministic system T . The following
	is well known result (see for instance [7, 6])
	Proposition 2.2. Let T satisfy assumptions (1)-(5). The following holds:
	(i) L φ : H loc α → H loc α has a dominant simple eigenvalue λ and its corre-
	sponding eigenfunction, ρ, is strictly positive.
	Let φ ∈ H loc α . The transfer operator, associated with T and with potential φ, acting on H loc α is defined as
	L φ (f )(x) :=	e φ(y) f (y).
	T y=x
	Let σ(L φ ) denote the spectrum of L φ as an operator on H loc α , and let L * φ
	denote the dual operator of L φ .

The justification and the interpretation of this condition will be discussed below in this section. See subsection 5.2.

This is equivalent to the condition θ({0}) > 0 in the continuous noise case.

In general one can replace the condition H K = ∅ by asking K i=1 p i 1 Xω i (x) > 0 and Proposition

[START_REF] Baladi | Positive transfer operators and decay of correlations[END_REF].1 would still hold. We asked explicitly for H K = ∅ to be one of the events because this is the most interesting case in our work.

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

Marseille where this work was initiated and finished. WB, JS and SV would like to thank the Isaac Newton Institute for Mathematical Sciences where

Example 6.2. We consider the following example: T (x) = 2x mod 1. Let H = (0, 1/2) be a hole. We now consider random holes in the following way: at each time n, H is either 'closed' with probability p, or H is 'open' with probability 1p . When acting on piecewise constant functions with respect to the partition [0, 1/2), [1/2, 1), the transfer operator L φ , with φ(x) =ln |T ′ x|, can be represented by the following matrix (acting by multiplication from the left):

The dominant eigenvalue of the above matrix 6 is given by

The corresponding left and right eigenvectors are, respectively,

Consequently, for any measurable set A, the absolutely continuous conditionally invariant measure α is given by:

To find the measure μφ,θ , we consider  * φ , the transpose of the matrix defined in (6 •10). Let Q be the stochastic matrix defined by

whose stationary probability vector q is defined by

Then μφ,θ , defined on cylinder sets, is given by μφ,θ (Z(j, a 0 , a 1 , . . . , a n-1

with μφ,θ (Z(j, a 0 )) = q a 0 , 6 The dominant eigenvalue, and the corresponding eigenfunction, of L φ when acting on Hölder functions, is the same as those of the above matrix reprsentation. The same is true for the dual operator L φ * using the above matrix with multiplication from the right.