Nicolai Haydn
email: nhaydn@math.usc.edu

Matthew Nicol
email: nicol@math.uh.edu

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

Licheng Zhang

Central limit theorems for the shrinking target problem

Suppose B i := B(p, r i ) are nested balls of radius r i about a point p in a dynamical system (T, X, µ). The question of whether T i x ∈ B i infinitely often ( i. o.) for µ a.e. x is often called the shrinking target problem. In many dynamical settings it has been shown that if E n := n i=1 µ(B i ) diverges then there is a quantitative rate of entry and lim n→∞ 1 En n j=1 1 B i (T i x) → 1 for µ a.e. x ∈ X. This is a self-norming type of strong law of large num-

bers. We establish self-norming central limit theorems (CLT) of the form lim n→∞ 1 an n i=1 [1 B i (T i x)µ(B i )] → N (0, 1) (in distribution) for a variety of hyperbolic and non-uniformly hyperbolic dynamical systems, the normalization constants are a 2 n ∼ E[ n i=1 1 B i (T i x)µ(B i )] 2 . Dynamical systems to

Introduction

Suppose (T, X, µ) is an ergodic dynamical system and B i (p) is a nested sequence of balls about a point p ∈ X. Recently there have been many papers concerning the behavior of the almost sure limit of the normalized sum 1 En n i=1 1 B i (p) (x) where [START_REF] Chernov | Dynamical Borel-Cantelli lemmas for Gibbs measures[END_REF][START_REF] Kim | The dynamical Borel-Cantelli lemma for interval maps[END_REF][START_REF] Galatolo | Skew products, quantitative recurrence, shrinking targets and decay of correlations[END_REF][START_REF] Gouëzel | A Borel-Cantelli lemma for intermittent interval maps[END_REF][START_REF] Gupta | A Borel-Cantelli lemma for non-uniformly expanding dynamical systems[END_REF][START_REF] Haydn | A note on Borel-Cantelli lemmas for non-uniformly hyperbolic dynamical systems[END_REF][START_REF] Jaerisch | A Frécehet law and an Erdös-Philipp law for maximal cuspidal windings[END_REF]. If the limit is known to exist almost surely then {B i (p)} is said to satisfy the Strong Borel Cantelli property.

E n := n i=1 µ(B i (p)) diverges
Many of the references we mentioned consider more general sequences of sets than nested balls. The study of hitting time statistics to a sequence of nested balls is sometimes called the shrinking target problem. In this paper we study self-norming central limit theorems for the shrinking target problem, namely the distribution limit of 1 an n i=1 [1 B i (p) -E n ] where a n is a sequence of norming constants. For reasons of exposition we focus on the case where µ(B i (p)) = 1 i , a critical case, where E n = log n. Our results extend (with obvious modifications to the norming sequences) to balls satisfying C 1 i γ 1 ≤ µ(B i (p)) ≤ C 2 i γ 2 where C 1 , C 2 are positive constants and 0 < γ 2 ≤ γ 1 ≤ 1. The main difficulty is to establish that the non-stationary variance has a limit in probability. Our results are limited to non-uniformly expanding systems i.e. those without a contracting direction and are based upon the Gordin [START_REF] Gordin | The central limit theorem for stationary processes[END_REF] martingale approximation approach (see also [START_REF] Liverani | Central limit theorem for deterministic systems[END_REF]). More generally, this paper is also an attempt to study the statistics of nonstationary stochastic processes arising as observations (which perhaps change over time) on an underlying dynamical system (which may change over time). Conze and Raugi [START_REF] Conze | Limit theorems for sequential expanding dynamical systems on [0,1[END_REF] studied similar problems for sequential expanding dynamical systems.

Somewhat related results were obtained by Nándori, Szász and Varjú [START_REF] Nándori | A central limit theorem for time-dependent dynamical systems[END_REF] who obtained central limit theorems in the setting in which a fixed observation φ : X → R was considered on a space on which a sequence of different transformations acted T i : X → X act, preserving a common invariant measure µ. The main difficulty in [START_REF] Nándori | A central limit theorem for time-dependent dynamical systems[END_REF] was also controlling the variance, but the setting in which the underlying maps change but the observation is fixed is simpler in some respects and more difficult in others.

We obtain fairly complete results in the case in which the transfer operator with respect to the invariant measure is quasicompact in the bounded variation norm.

These results are contained in Proposition 5.1 and Theorem 6.4. For systems in which the transfer operator is quasicompact in a Hölder or Lipschitz space we show that under the assumption we call (SP) (derived from a Gal-Koksma lemma as formulated by Sprindzuk [START_REF] Vladimir | Metric theory of Diophantine approximations[END_REF]) or a form of short returns assumption called Assumption C we have a central limit theorem ( Theorem 3.1). Assumption C and the SP property have been shown to hold for generic points in a variety of non-uniformly expanding systems [START_REF] Collet | Statistics of closest return for some non-uniformly hyperbolic systems[END_REF][START_REF] Gupta | Extreme value theory and return time statistics for dispersing billiard maps and flows, Lozi maps and Lorenz-like maps[END_REF][START_REF] Holland | Extreme value distributions for nonuniformly hyperbolic dynamical systems[END_REF].

In Section 2 we discuss the set-up, describe the martingale approach we use, prove some general results on variance and discuss the SP property and Assumption C. Section 3 gives our results under the assumption of quasi-compactness in Hölder norms and also some applications. In Section 4 we give our results when we have quasi-compactness of the transfer operator in the bounded variation norm, and we give applications to piecewise expanding maps in higher dimensions. The last section is a concluding discussion, while the Appendices describe the Gal-Koksma lemma we use and show that Assumption C is satisfied for generic points in many of our applications.

The setup.

We suppose that (T, X, µ) is an ergodic dynamical system. Let the transfer operator P be defined by φψ•T dµ = P φψdµ for all φ, ψ ∈ L 2 (µ) so that P is the adjoint of the Koopman operator Uφ := φ • T with respect to the invariant measure µ. Suppose B α is a Banach space of functions and φ 1 ≤ C φ α where . α is the Banach space norm and . 1 is the L 1 norm with respect to µ. We assume P restricts to an operator

P : B α → B α such that P n φ α ≤ C 1 θ n φ α for all φ ∈ B α such that φ dµ = 0.
This implies exponential decay of correlations of the form, that for some 0 < θ < 1,

| φψ • T n dµ -( φ dµ)( ψ dµ)| ≤ Cθ n φ α ψ 1
for all φ ∈ B α , ψ ∈ L 1 µ. In our applications we will have the pairs (BV (X), L 1 (µ))

or (H γ (X), L 1 (X)) where BV (X) is the space of function of bounded variation and H γ (X) is the space of Hölder functions of exponent γ. For example if T is a smooth uniformly expanding map of the unit interval X then B α could be taken as the Banach space of functions of bounded variation BV (X). In this paper we will consider Lipschitz rather than Hölder functions, as our results and proofs immediately generalize to the Hölder setting with the obvious changes.

Remark 2.1 The weaker assumption of exponential decay of correlations

| φψ • T n dµ -( φ dµ)( ψ dµ)| ≤ Cθ n φ α ψ ∞
implies that P n φ 1 ≤ Cθ n φ α (by taking ψ to be sign(P n φ)) and hence P contracts exponentially in the L 1 norm. This assumption is sufficient for all our results on variance in Section 2, with the exception of the proof of the boundedness of the terms w j , given in Lemma 2.9 which seems to require our stronger assumption that P n φ α ≤ C 1 θ n φ α . These estimates on the growth of w j are used in the proof of Theorem 3.1. If B α is the space of functions of bounded variation then the w j terms are easily seen to be uniformly bounded under the assumption P n φ BV ≤ Cθ n φ BV .

Let p ∈ X and let B n (p) be a sequence of nested balls about p such that µ(B n (p)) = 1 n . Let 1 Bn(p) be the characteristic function of B n (p). We will sometimes write E[φ] or φ for the integral φ dµ when the context is understood. Our results generalize immediately to sequences of nested balls with bounds We define φ α n = φα n -φα n so that φ α n = 0. For ease of notation we will subsequently drop the superscript α on φ α n and φα n . Define φ 0 = 0 and for n ≥ 1

C 1 n γ 1 ≤ µ(B n (p)) ≤ C 2 n γ 2 for constants C 1 , C 2 > 0 and 0 < γ 2 ≤ γ 1 ≤ 1 (
w n = P φ n-1 + P 2 φ n-2 + . . . + P n φ 0 = n j=1
P j φ n-j so that w 1 = P φ 0 , w 2 = P φ 1 + P 2 φ 0 , w 3 = P φ 2 + P 2 φ 1 + P 3 φ 0 etc... For n ≥ 1 define

ψ n = φ n -w n+1 • T + w n
Recall our assumptions φn α ≤ Cn k (so φ n α ≤ Cn k ) and

P n φ α ≤ C 1 θ n φ α for all φ ∈ B α such that φdµ = 0. Hence w n α ≤ C 2 φ n α , w n • T α ≤ C 3 φ n α
(since UP φ α ≤ C φ α for all φ ∈ B α ) and hence ψ n α ≤ C 4 φ n α . Using the fact that P (w n+1 • T ) = w n+1 P 1 = w n+1 one may show that P ψ n = 0.

Since UP (•) = E[•|T -1 B], P ψ j = 0 implies that E[ψ j |T -1 B] = 0 and in turn E[ψ j • T j |T -1-j B] = 0 (since T preserves µ). Furthermore ψ j • T j is T -j B measurable for all j ≥ 0.
Following the approach of Gordin we will express n j=1 φ j • T j as the sum of a (non-stationary) martingale difference array and a controllable error term and then use the following Theorem 3.2 from Hall and Heyde [START_REF] Hall | Martingale limit theory and its application, Probability and Mathematical Statistics[END_REF]:

Theorem 2.3 (Theorem 3.2 [16]) Let {S n,i , F n,i , 1 ≤ i ≤ k n ,
n ≥ 1} be a zeromean square-integrable martingale difference array with differences X n,i and let η 2 be an almost sure finite random variable. Suppose that:

(a) max i |X n,i | → 0 in probability; (b) i X 2 n,i → η 2 in probability; (c) E(max i X 2 n,i ) is bounded in n; (d) the σ-fields are nested: F n,i ⊂ F n+1,i for 1 ≤ i ≤ k n , n > 1.
Then S n,kn → Z (in distribution) where the random variable Z has the character-

istic function E(exp(-1 2 η 2 t 2 )).
As is common in the application of martingale theory to non-invertible dynamical systems we will have to consider the natural extension so that we have a martingale in backwards time. We outline our scheme of proof.

Let (σ, Ω, m) be the natural extension of (T, X, µ). Each ψ j lifts to to a function ψ * j on Ω in a natural way, ψ * j (. . . ω -2 ω -1 .ω 0 ω 1 . . .) := ψ j (ω 0 ). To simplify notation we write simply ψ j instead of ψ * j . We define scaling constants by a 2 n = E( n j=1 φ j • T j ) 2 . This sequence of constants play the role of non-stationary variance. Giving estimates on the growth and nondegeneracy of a n in this non-stationary setting is more difficult than in the usual stationary case.

We define a triangular array X n,i = 1 an ψ n-i • σ -i , i = 1, . . . , n, n ∈ N, and put S n,i = i j=1 X n,j for the partial sums (along rows). Then X n,i is F i := σ i B 0 measurable where B 0 is the σ-algebra B lifted to Ω. Note that in Theorem 2.3 we take F n,i := F i for all n and k n = n. The F i form an increasing sequence of σ-algebras. We obtain

E[S n,i+1 |F i ] = S n,i + E[X n,i+1 |F i ] where by stationarity E[X n,i+1 |F i ] = E[ψ n-i-1 |σ -1 B 0 ] = 0. Hence E[S n,i+1 |F i ] = S n,i
[w j • T j -w j • T j+1 ] → 0 in L 1 which implies that lim n→∞ 1 an n-1 j=0 φ j •T j → N(0, 1) in distribution.

Some lemmas on variance

In this section we establish some preliminary results on the growth of the variance E[( n j=1 φ j ) 2 ] that will be useful in determining the scaling constants a n . For further reference let us notice that P n φ α ≤ C 1 θ n φ α and φ 1 ≤ C 1 φ α and that there exists a constant a such that

j>a log i P j φ i 1 ≤ 1 i 3 . (2.1) Lemma 2.4 lim sup n→∞ 1 log n E( n i=1 φ i • T i ) 2 ≥ 1
Proof: By exponential decay of correlations and (2.1) we get for the long term interactions:

j>a log i+i φ i • T i φ j • T j ≤ c 1 i 2 ,
where we used exponential decay and our bound φ j 1 ≤ C 1 φ j α ≤ Cj k , where C,k are independent of j. This bound is from assumption (ii). Recall φ j = φjφj and φj 1 ≤ C 3 j (for some C 3 ). Thus for the short term interactions we get

i+a log i j=i+1 φ i • T i φ j • T j = i+a log i j=i+1 φi • T i φj • T j + O( a log i i 2 ) whence n i=1 j>i E[φ i • T i φ j • T j ] = O(1) + n i=1 i+a log i j=i+1 E[ φi • T i φj • T j ]. Since E( n i=1 φ i • T i ) 2 = n i=1 E(φ 2 i ) + 2 n i=1 j>i E[φ i • T i φ j • T j ] and n i=1 i+a log i j=i+1 E[ φi • T i φj • T j ] ≥ 0 the lemma is proved. Lemma 2.5 n i=1 n j=i+1 φ i • T i φ j • T j = n i=1 (φ i w i ) • T i
Proof: Recalling that φ 0 = 0 this follows by a direct calculation and rearrangement of terms as

n-1 i=1 n j=i+1 φ i • T i φ j • T j = n j=2 j-1 i=1 φ i • T i φ j • T j = n j=2 j-1 i=1 P j-i φ i φ j = n j=2 ( j-1 i=1 P j-i φ i )φ j = n j=2 w j φ j .
The following lemma is the main result of this subsection:

Lemma 2.6

a n = E( n i=1 φ i • T i ) 2 = n i=1 E[ψ 2 i ] -w 2 1 + w 2 n+1
Proof: Let us first observe that factoring out yields

ψ 2 j = φ 2 j + 2φ j (w j -w j+1 • T ) + (w j -w j+1 • T ) 2 = φ 2 j + 2φ j (w j -w j+1 • T ) + w 2 j + w 2 j+1 • T -2w j w j+1 • T
which when integrated leads to

ψ 2 j = φ 2 j + 2 φ j (w j -w j+1 • T ) + w 2 j + w 2 j+1 -2 w j w j+1 • T = φ 2 j + 2 φ j w j -2 P φ j w j+1 + w 2 j + w 2 j+1 -2 P w j w j+1 = φ 2 j + 2 φ j w j -2 P φ j w j+1 + w 2 j + w 2 j+1 -2 (w j+1 -P φ j )w j+1 = φ 2 j + 2 φ j w j + w 2 j -w 2 j+1 .
Since by Lemma 2.5

a n = n i=1 E(φ 2 i ) + 2 n i=1 n j=i+1 φ i • T i φ j • T j = n i=1 E(φ 2 i ) + 2 (φ i w i ) • T i
the statement follows by substituting ψ 2 j -w 2 j + w 2 j+1 for the terms inside the sum on the RHS and then telescoping out the expected values of w 2 j .

Property (SP)

Several authors [START_REF] Kleinbock | Logarithm laws for flows on homogeneous spaces[END_REF][START_REF] Chernov | Dynamical Borel-Cantelli lemmas for Gibbs measures[END_REF] have used a property derived from the Gal-Koksma theorem (see Appendix) to prove the SBC property for sequences of balls. Later we will show that in certain settings the (SP) property also implies a CLT.

Suppose B i are balls and let

f i = 1 B i • T i . If n i=m n j=i+1 E(f i f j ) -E(f i )E(f j ) ≤ C n i=m E(f i ) (SP )
for arbitrary integers n > m then the balls are said to have the (SP) property.

Short returns and Assumption (C)

In this section we discuss a condition on short return times first considered, to our knowledge, by P. Collet [START_REF] Collet | Statistics of closest return for some non-uniformly hyperbolic systems[END_REF]. We have called it Assumption (C). This condition has been used to establish extreme value statistics [START_REF] Collet | Statistics of closest return for some non-uniformly hyperbolic systems[END_REF][START_REF] Holland | Extreme value distributions for nonuniformly hyperbolic dynamical systems[END_REF][START_REF] Gupta | Extreme value theory and return time statistics for dispersing billiard maps and flows, Lozi maps and Lorenz-like maps[END_REF] and dynamical Borel-Cantelli lemmas [START_REF] Gupta | A Borel-Cantelli lemma for non-uniformly expanding dynamical systems[END_REF][START_REF] Haydn | A note on Borel-Cantelli lemmas for non-uniformly hyperbolic dynamical systems[END_REF].

Suppose p ∈ X and B i (p) is a nested sequence of balls centered at a point p, with lim i µ(B i (p)) = 0.

Assumption (C):

We say (B i (p)) satisfies assumption (C) if there exists η(p) ∈ (0, 1) and κ(p) > 1 such that for all i sufficiently large

µ(B i (p) ∩ T -r B i (p)) ≤ µ(B i (p)) 1+η
for all r = 1, . . . , log κ i.

If (B i (p)) satisfies assumption (C) then we can say more about the behavior of the constants a n .

Lemma 2.7 Under Assumption (C) there exists a constant C 1 so that

|φ j w j | ≤ C 1 log j j 1+η .
Proof: By the contraction property of the transfer operator one has by (2.1) for a sufficiently large constant a i<j-a log j

φ j P j-i φ i ≤ 1 j 2 .
Let φ j = φjφj where φj is the B α approximation to 1 B j (p) and note that φj 1 , φ j 1 ≤ c 1 j (for some c 1 ). Hence we obtain in the L 1 -norm: (as φj ≥ 0)

|φ j w j | ≤ a log j n=1 φj P n φj-n + φ2 j-n + φj P n φj-n + φj φj-n + O 1 j 2 = a log j n=1 φj P n φj-n + 3µ( φj-n ) 2 + O 1 j 2 = a log j n=1 φj P n φj-n + O log j j 2 ,
where we used that φj = O(j -1 ). Now by assumption (C) we have

φj P n φj-n ≤ φj-n • T n φj-n ≤ µ(B j-n ∩ T -n B j-n ) ≤ c 2 (j -n) 1+η ,
for n ≤ a log j, and thus a log j n=1 φj P n φj-n ≤ c 3 j 1+η a log j, proving the lemma.

Lemma 2.8 If (B i (p)) satisfies Assumption (C) then E( n i=1 φ i • T i ) 2 = n i=1 E[φ 2 i ] + O(1) = log n + O(1)
.

Proof: Rearranging the sums yields by Lemma 2.5

E( n i=1 φ i • T i ) 2 = n i=1 E[φ 2 i ] + 2 n-1 i=1 n j=i+1 φ i • T i φ j • T j = n i=1 E[φ 2 i ] + 2 n j=2 w j φ j
and hence the result follows by Lemma 2.7 as η > 1

Bounds on w j

We now assume that φ ∞ ≤ C φ α which under our assumption on the transfer operator implies that for a mean-zero function φ ∈ B, P n φ ∞ ≤ Cθ n φ α for some C, 0 < θ < 1 independently of φ. For example if . α were the Banach space of Hölder functions of exponent γ on the unit interval then φ ∞ ≤ C φ α . In the BV or quasi-Hölder norm indicator functions are bounded, and the proof that w j is uniformly bounded is straightforward in this case.

Lemma 2.9 Assume P n φ ∞ ≤ Cθ n φ α then there exist a constant C 2 such that w j ∞ < C 2 for all j where E n = n j=1 µ(B i ).

Proof: For some a > 0 we can achieve

n j=a log n |P j φ n-j | ∞ ≤ c 1 n j=⌊a log n⌋ θ j (n - j) k = O(n -2 ) and in particular |P j φ n-j | ∞ = O(n -2
) for all j ≥ a log n and all n. As in the previous lemma let φj be the B α approximation for 1 B j and φ j = φj -µ( φj ). In view of the tail estimate it is only necessary to bound ⌊a log n⌋ j=1 P j φ n-j independently of n.

(i) Bound from below: Since φ j ≥ -µ( φj ) = O(j -1 ) one obtains ⌊a log n⌋ j=1

P j φ n-j ≥ ⌊a log n⌋ j=1 c 2 (n -j) -γ 1 ≥ -c 3 log n n γ 1
for some constants c 2 , c 3 independent of j and n. Hence w n ≥ -c 4 for some c 4 > 0 and all n (independent of γ1, γ 2 ).

(ii) Bound from above: Since 1 B j+1 ≤ 1 B j one has φj+1 ≤ φj and in particular µ( φj+1 ) ≤ µ( φj ). Hence φ j+1 -φ j ≤ µ( φj ) -µ( φj+1 ) and (as φ 0 = 0)

w m -w m-1 = m-1 j=1 P j (φ m-j -φ n-1-j ) + P m φ 0 ≤ m-1 j=1 µ( φm-1-j ) -µ( φm-j ) ≤ ⌊a log m⌋ j=1 µ( φm-1-j ) -µ( φm-j ) + O(m -2 ).
Consequently (w 1 = P φ 0 = 0)

w n = n m=2 (w m -w m-1 ) + w 1 ≤ n m=2   ⌊a log m⌋ j=1 µ( φm-1-j ) -µ( φm-j ) + O(m -2 )   = ⌊a log n⌋ j=1 n m=2∨⌈e j a ⌉ µ( φm-1-j ) -µ( φm-j ) + O(m -2 ) = ⌊a log n⌋ j=1 µ( φ2∨⌈e j a ⌉-j ) -µ( φn-j ) + O((2 ∨ e j a ) -1 ) ≤ C 3 for a constant C 3 independent of n because ⌊a log n⌋ j=1 µ( φn-j ) ≤ c 5 a log n n γ 2 → 0 as n → ∞ and ⌊a log n⌋ j=1 µ( φ2∨⌈e j a ⌉-j ) ≤ c 6 ⌊a log n⌋ j=1 (e j a ) -γ 2 = O(1)
for constants c 5 , c 6 independent of n.

3 Decay in Lipschitz versus L 1

We take B α to be the space of Lipschitz functions, the arguments we give hold for Hölder norms with obvious modification. We assume that the transfer operator P , when restricted to Lip(X), contracts exponentially:

||P n φ|| Lip ≤ Cθ n ||φ|| Lip (3.1)
for all Lipschitz functions φ such that φ dµ = 0 for some θ ∈ (0, 1), where θ and C independent of φ.

This implies

φψ • T n dµ -E[φ]E[ψ] ≤ Cθ n φ Lip ψ L 1 (3.2)
for the same θ ∈ (0, 1) and C independent of φ, ψ.

For a sequence of (nested) balls B i we put E n = n i=1 µ(B i ) and S n = n i=1 1 B i •T i for the 'hit counter' for an orbit segment of length n. The sequence of balls B i satisfies the strong Borel-Cantelli (SBC) property if

lim n→∞ S n (x) E n = 1 (3.3)
for almost every x ∈ X.

Theorem 3.1 Assume that the transfer operator, when restricted to Lip(X), contracts exponentially as in (3.1) for some θ ∈ (0, 1).

Suppose B i (p) be nested balls about a point p with µ(B i ) = 1 i . Let a 2 n = E( n j=1 (1 B i -1 i )) 2 .
(I) If the nested sequence of balls (B i (p)) satisfies Assumption (C) and the SBC property (3.3) then

a 2 n = log n + O(1)
and 1 √ log n n j=1 1 B i - 1 i → N(0, 1)
in distribution.

(II) If (B i (p)) has the SP property then

1 √ a n n j=1 (1 B i • T i - 1 i ) → N(0, 1).
Proof: We will let φ j = φjφj ,where φj be a Lipschitz approximation to 1

B j , such that        φj -1 B j 1 < 1 j 2 φj Lip ≤ Cj k φj ≥ 0 .
We define w n = P φ n-1 + P 2 φ n-2 + . . . + P n φ 0 and put ψ n = φ n -w n+1 • T + w n .

Then P ψ n = P φ n -w n+1 + n j=2 P j φ n-j+1 = 0 which corresponds to ψ n χ • T dµ = χP ψ n dµ = 0 for any integrable χ. Note that

||φ j || ∞ ≤ ||φ j || Lip , ||φ j || 1 ≤ ||φ j || Lip . Lemma 3.2 There exist constants C 4 , k, a so that (I) w n Lip ≤ C 4 n k , (II) w n ∞ ≤ C 4 , (III) w n 1 ≤ C 4 log n n .
Proof of Lemma 3.2. (I) By the contraction of the transfer operator for Lipschitz continuous functions one obtains

w n Lip ≤ ∞ j=0 P j φ n Lip ≤ ∞ j=0 C 1 θ j φ n Lip ≤ c 1 n k
(II) Is a consequence of Lemma 2.9.

(III) For sufficiently large a we get

||w n || 1 ≤ a log n j=1 ||P j φ n-j || 1 + n j=a log n+1 ||P j φ n-j || 1 ≤ a log n j=1 ||φ n-j || 1 + ∞ j=a log n+1 ||P j φ n-j || Lip ≤ a log n j=1 ||φ n-j || 1 + n j=a log n+1 C 1 θ j ||φ n-j || Lip ≤ a log n n -a log n + c 4 log 2 n n 2 ≤ c 5 log n n
for some c 4 , c 5 independent of n.

Now put C 4 = max(c 1 , c 5 ).
As before let (σ, Ω, m) be the natural extension of (T, X, µ) and put a 2 n = E( n j=1 φ j • T j ) 2 for the rescaling factors where the ψ j lift to Ω in a natural way. By Assumption (C), a 2 n ∼ log n by Lemma 2.8. Again we put X n,i = 1 an ψ n-i • σ -i , i = 1, . . . , n which are F i = σ i B 0 measurable where B 0 is the σ-algebra B lifted to Ω. The F i form an increasing sequence of σ-algebras. We put S n,i = i j=1 X n,j , i = 1, . . . , n (k n = n), where the X n,i and obtain

E[S n,i+1 |F i ] = S n,i + E[X n,i+1 |F i ] but by sta- tionarity E[X n,i+1 |F i ] = E[φ n-i-1 |σ -1 B] = 0. Hence E[S n,i+1 |F i ] = S n,i
and X n,i is a martingale difference array with respect to F i .

We now show condition (a) and (c) hold (clearly (d) holds). To see (a) and (c)

calculate ψ 2 n dµ ≤ ψ n ∞ ψ n 1 ≤ C log 2 n n .
Hence condition (a) and (c) hold. We now prove (I) and show that under Assumption (C), n i=1 X 2 n,i → 1 in probability and hence condition (b) holds.

Lemma 3.3 1 log n n j=1 ψ 2 j • T j → 1 in probability as n → ∞.
Proof. We follow an argument given by Peligrad [START_REF] Peligrad | Central limit theorem for triangular arrays of non-homogeneous Markov chains To appear in Prob. Theory and Related Fields[END_REF]. As ψ j = φ j + w j -w j+1 • T we obtain

ψ 2 j = φ 2 j + 2φ j w j + w 2 j + w 2 j+1 • T -2w j+1 • T (φ j + w j ) = (φ 2 j + 2φ j w j + w 2 j + w 2 j+1 • T -2w j+1 • T (ψ j + w j+1 • T ) = φ 2 j + (w 2 j -w 2 j+1 • T ) -2ψ j w j+1 • T + 2φ j w j .
We want to sum over j = 1, . . . , n and normalize by log n and wish to estimate the error terms which are the last four terms on the RHS. The terms w 2 j -w 2 j+1 • T are bounded and telescope so may be neglected.

In order to estimate the third of the error terms, ψ j w j+1 • T we proceed like Peligrad (page 9) using a truncation argument. Let w ǫ j = w j 1 {|w j |≤ǫ √ log n} , where for simplicity of notation we have left out the dependence on n. Then

n j=1 ψ j • T j w ǫ j+1 • T j+1 2 = n j=1 ψ j • T j w ǫ j+1 • T j+1 2 ≤ ǫ 2 n j=1 ψ 2 j
since the cross terms vanish (for j > i), as

(ψ j w ǫ j+1 • T ) • T j (ψ i w ǫ i+1 • T ) • T i = (ψ j w ǫ j+1 • T ) • T j-i (ψ i w ǫ i+1 • T ) = (ψ j w ǫ j • T ) • T j-i-1 P (ψ i w ǫ i • T ) = (ψ j w ǫ j+1 • T ) • T j-i-1 w ǫ i+1 P ψ i = 0 as P (ψ i w ǫ i+1 • T ) = w ǫ i+1 P ψ i .
For any a > ǫ we obtain using Tchebycheff's inequality (on the second term):

P 1 log n n j=1 ψ j • T j w j+1 • T j+1 > a ≤ P max 1≤j≤n 1 √ log n w j+1 • T j+1 > ǫ + P 1 log n n j=1 ψ j • T j w ǫ j+1 • T j+1 > a ≤ P ( max 1≤j≤n |w j+1 • T j+1 | > ǫ log n) + ǫ 2 a 2 log n n j=1 ψ 2 j = P ( max 1≤j≤n |w j • T j+1 | > ǫ log n) + c 1 ǫ 2 a 2 .
In the last line we used n j=1 E[ψ 2 j ] ∼ log n by Lemma 2.6 and Lemma 2.8. By boundedness of the w j (Lemma 2.9) one gets that P (max

1≤j≤n |w j+1 • T j+1 | > ǫ √ log n) → 0 for every ǫ > 0 as n → ∞. Choosing a = ǫ 1 2
we conclude that 1 an n j=1 ψ j • T j w j+1 • T j+1 converges to zero in probability as n → ∞. For the fourth error term 1 log n 2 n j=1 (φ j w j ) • T j we obtain by Lemma 2.8: we proceed as in the proof of (I) except for the verification of condition (b). We will prove a SBC property for φ 2 j +2w j φ j . Decomposing φ j = φj -µ( φj ) and defining wj = P φj-1 +...+P [a log j] φj-[a log j] we see that φ 2 j -φ2 j 1 ≤ C j 2 and w j φ j -wj φj 1 ≤ C log j φ2 j • T j + 2( wj φj ) • T j → 1 almost surely. For this we want to use Proposition 8.1 with f j = φ2 j + 2 wj φj , g j = f j and h j to be determined below. We need to estimate the terms in

n j=1 (φ j w j ) • T j 1 ≤ n j=1 φ j w j 1 ≤ c 2 ∞ j=1 log j j 1+η < ∞ uniformly in n.
n i=m φ2 j + 2 wj φj 2
In order to verify the condition of the proposition we look at the three individual sums as follows:

(i) The fact that condition (SP) holds for the functions φj implies

n i=m n j=i+1 φj • T j-i ( φi ) -E[ φj ]E[ φi ] ≤ C n i=m E[ φj ]. Since E( φ2 j ) -E( φj ) = O(j -k ) we obtain n i=m n j=i+1 φ2 j • T j-i φ2 i -E[ φ2 j ]E[ φ2 i ] ≤ C n i=m E[ φj ] + n i=m O(i -k+1 ). (ii) Lemma 3.2 (| wj | ∞ ≤ C 4 ∀j) now yields n i=m n j=i+1 ( φj wj ) • T j-i ( φi wi ) -E[ φj wj ]E[ φi wi ] ≤ CC 2 4 n i=m E[ φj ] + n i=m O(i -k+1
).

(iii) In the same way we obtain for the 'mixed' terms

n i=m n j=i+1 ( φj wj ) • T j-i φ2 i -E[ φj wj ]E[ φ2 i ] ≤ CC 4 n i=m E[ φj ] + n i=m O(i -k+1 ) .
Combining (i), (ii) and (iii) yields for all m < n and some constant c 1 :

n i=m φ2 j + 2 wj φj 2 ≤ c 1 n i=m E( φj ) + O(i -k+1 )
which by Proposition 8.1 implies that 1

En n j=1

φ2 j • T j + 2( wj φj ) • T j → 1 almost surely, provided k ≥ 2.
4 Applications to dynamical systems.

Theorem 3.1 applies to a variety of dynamical systems including Gibbs-Markov maps [START_REF] Aaronson | Denker Local Limit Theorems for Gibbs-Markov Maps[END_REF] and rational maps [START_REF] Haydn | Convergence of the transfer operator for rational maps[END_REF]. For Gibbs-Markov maps it has been shown [14, Theorem 1] that nested sequences of balls (B i (p)) satisfy both the Strong Borel Cantelli property and assumption C, so that (I) applies. For rational maps [START_REF] Haydn | Convergence of the transfer operator for rational maps[END_REF]Theorem 10] shows that the transfer operator contracts exponentially in the L ∞ norm hence if the (SP) property is also proved then (II) holds. More generally (II) shows that proving the (SP) property for systems whose associated transfer operator has exponential decay suffices to prove the SBC property and the CLT for shrinking targets.

Decay in BV (X) versus L 1

It is known that summable decay of correlations in BV (X) versus L 1 implies the SP property by work of Kim [25, Proof of Theorem 2.1] (see also Gupta et al [START_REF] Gupta | A Borel-Cantelli lemma for non-uniformly expanding dynamical systems[END_REF]Proposition 2.6]). Hence the statement in this setting is simpler.

Let the transfer operator P be defined by φψ • T dµ = P φψ dµ for all φ, ψ ∈ L 2 (µ), that is P is the adjoint of the Koopman operator Uφ := φ • T .

We assume that the restriction of P to the space BV (X) is exponentially contracting, i.e. P : BV (X) → BV (X) satisfies

P n φ BV ≤ Cθ n φ BV (5.1)
for all φ ∈ BV (X) such that φ dµ = 0.

This implies that (T, X, µ) has exponential decay of correlations in BV versus L 1 , so that for some 0 < θ < 1,

φψ • T n dµ -( φ dµ)( ψ dµ) ≤ Cθ n φ BV ψ 1 (5.2)
for all φ ∈ BV (X), ψ ∈ L 1 (µ). In particular the measure µ is ergodic.

Proposition 5.1 Assume the transfer operator P contracts exponentially as given by (5.1)

Let B i := B(p, r i ) be nested balls of radius r i about a point p such that µ(B i ) = 1 i , and

a 2 n = E( n j=1 (1 B i • T i -1 i )) 2 . Then: (I) lim sup n→∞ an √ log n ≥ 1 and 1 a n n j=1 (1 B i • T i - 1 i ) → N(0, 1).
(II) If the nested sequence of balls (B i (p)) about p satisfies Assumption (C) then

a 2 n = E[( n j=1 (1 B i • T i - 1 i )) 2 ] = log n + O(1)
and

1 √ log n n j=1 (1 B i • T i - 1 i ) → N(0, 1)
in distribution.

Proof: The proof is the same as for Theorem 3.1 with the simplification that the SP property holds automatically as we have summable decay of correlations in BV (X)

versus L 1 (see proof of [25, Theorem 2.1]). Furthermore Lemma 2.4 shows that the variance is unbounded and Lemma 2.8 gives a precise rate of growth in the case that Assumption (C) holds. 1 Bn (p). The rates of growth are given by Lemma 2.4 which shows that the variance is unbounded. Lemma 2.8 gives a precise rate of growth in the case that Assumption (C) holds. In Proposition 6.2 we extend these results to piecewise expanding maps in higher dimensions.

6 Applications of Proposition 5.1.

Proposition 5.1 applies to certain classes of one-dimensional maps such as piecewise expanding maps of the interval T : X → X with 1 T ′ of bounded variation and possessing an absolutely continuous invariant measure with density bounded away from zero (those maps satisfying the assumptions of [25, Theorem 2.1], see also [START_REF] Gupta | A Borel-Cantelli lemma for non-uniformly expanding dynamical systems[END_REF]). For these systems, Assumption (C) has been shown to hold for nested balls about µ a.e. p ∈ X [START_REF] Holland | Extreme value distributions for nonuniformly hyperbolic dynamical systems[END_REF][START_REF] Gupta | Extreme value theory and return time statistics for dispersing billiard maps and flows, Lozi maps and Lorenz-like maps[END_REF]. In the next subsection we generalize these results to piecewise expanding maps in higher dimensions.

Piecewise expanding maps in higher dimensions

In this section we prove the Strong Borel Cantelli property and the CLT for shrinking balls in a class of expanding maps in higher dimensions. We also show that assumption C holds for µ-a.e. point.

The Banach spaces will be given by L1 , defined with respect to the Lebesgue measure on R n , and a quasi-Hölder space with properties analogous to BV which we define below. A key property of the quasi-Hölder space is that characteristic functions of balls have bounded norm (as in the BV norm) which turns out to be a very useful property.

The maps are defined on compact sets Z ∈ R N . Denote by dist(•, •) the usual metric in R N and for ε > 0 let B ε (x) = {y ∈ R N : dist(x, y) < ε} be the ε-ball centred at x. Let B ε (A) = {y ∈ R N : dist(y, A) ≤ ε} and write Z • for the interior of Z and Z its closure.

A map T : Z → Z is said to be a multidimensional piecewise expanding map, if there exists a family of finitely many disjoint open sets {Z i } such that Leb(Z \ i Z i ) = 0 and there exist open sets Z i ⊃ Z i and C 1+α maps T i : Z i → R N (for some 0 < α ≤ 1) and some sufficiently small real number ε 1 > 0 such that for all i,

• (H1) T i ( Z i ) ⊃ B ε 1 (T (Z i )) and T i | Z i = T | Z i ; • (H2) For x, y ∈ T (Z i ) with dist(x, y) ≤ ε 1 , | det DT -1 i (x) -det DT -1 i (y)| ≤ c| det DT -1 i (x)|dist(x, y) α ;
• (H3) There exists s = s(f

) < 1 such that ∀x, y ∈ T ( Z i ) with dist(x, y) ≤ ε 1 , we have dist(T -1 i x, T -1 i y) ≤ s dist(x, y). • (H4) Let G(ε, ε 1 ) := sup x G(x, ε, ε 1 ) where G(x, ε, ε 1 ) := i Leb(T -1 i B ε (∂T Z i ) ∪ B (1-s)ε 1 (x)) Leb(B (1-s)ε 1 (x)) (6.1)
and assume that sup

δ≤ε 1 s α + 2 sup ε≤δ G(ε) ε α δ α < 1 1 (6.2)
We now introduce the Banach space of quasi-Hölder functions in which the spectrum of the Perron-Frobenius operator P is investigated. Given a Borel set Γ ⊂ Z, we define the oscillation of ϕ ∈ L 1 (Leb) over Γ as osc(ϕ, Γ) := ess sup

Γ ϕ -ess inf Γ ϕ.
The function x → osc(ϕ, B ε (x)) is measurable (see [?, Proposition 3.1]) For 0 < α ≤ 1 and ε 0 > 0, we define the α-seminorm of ϕ as

|ϕ| α = sup 0<ε≤ε 0 ε -α R N osc(ϕ, B ε (x)) dLeb(x).
Let us consider the space of functions with bounded α-seminorm

V α = {ϕ ∈ L 1 (Leb) : |ϕ| α < ∞},
and endow V α with the norm

• α = • 1 + | • | α
which makes it into a Banach space. We note that V α is independent of the choice of ε 0 and that V α is continuously injected in L ∞ (Leb). According to [?, Theorem 5.1], there exists an absolutely continuous invariant probability measure (a.c.i.p.) µ, with density bounded above, and bounded below from zero, which has exponential decay of correlations against L 1 observables on the finitely many mixing components of V α : in view of the next Theorem 6.4 we will from now restrict ourselves to one of those components, by taking a mixing iterate of T . More precisely, if the map T is as codimension one embedded compact submanifold, then define the quantity:

η 0 (T ) := s α + 4s 1 -s Y (T ) γ N -1 γ N
where

Y (T ) = sup x i # {smooth pieces intersecting ∂V i containing x} ,
is the maximal number of smooth components of the boundaries that can meet in one point and

γ N = π N/2
(N/2)! , the N -volume of the N -dimensional unit ball of R N . We require that η 0 (T ) < 1, and this may replace the condition (6.2) above. defined above and if µ is the mixing a.c.i.p., then there exist constants C < ∞ and

γ < 1 such that Z ψ • T n h dµ -ψdµ hdµ ≤ C ψ L 1 h α γ n (6.3)
for all ψ ∈ L 1 and for all h ∈ V α . Moreover P n φ α ≤ C φ α for all φ ∈ V α and thus equation 5.1 holds.

We now show that characteristic functions of balls are bounded in the • α norm.

Lemma 6.1 Let B i (p) be a nested sequence of balls about a point p ∈ X, then there exists a constant C 3 (α) such that

1 B i α ≤ C 3 (α)
for all i.

Proof: Take any set A with a rectifiable boundary. If p is not in a 2ǫ neighborhood of the boundary of A, then the oscillation is zero, otherwise it is 1. Therefore we have The boundedness of the characteristic functions in the • α -norm allows us to proceed as in Proposition 5.1 (see also [START_REF] Conze | Limit theorems for sequential expanding dynamical systems on [0,1[END_REF]) and to obtain the following result.

Proposition 6.2 Assume a piecewise expanding map T on a compact set Z ⊂ R n satisfies conditions (H1)-(H4) and is mixing with respect to its absolutely continuous invariant measure µ. Let B i := B(p, r i ) be nested balls of radius r i about a point p

such that µ(B i ) = 1 i . Then the variance a 2 n := E[( n j=1 (1 B i • T i -1 i )) 2 ] satisfies a n √ log n ≥ 1 and 1 √ a n n j=1 (1 B i • T i - 1 i ) → N(0, 1)
in distribution.

Proof: The SBC property (I) is immediate from the decay of correlations, Equation 6.3 and the bound 1 B i α ≤ C 3 (α) by the proof of Proposition 6.1. The growth estimate follows from Lemma 2.3.

We now make an additional assumption. Suppose that we have M domains of local injectivity for the map T ; if we take the join Z j := j-1 i=0 T -i Z, where Z denotes the partition, mod-0, into the closed sets Z In order to prove condition (C) we require a further assumption which is also called the finite range structure. We assume: j) . Then U consists of only finitely many subsets of Z with positive Lebesgue measure, hence U m = inf U ∈U m(U) is bounded below. We now fix j and consider the cylinder, say, Z (j) l . Let us suppose that {z k } k≥1 is a sequence of points in Z (j) l converging to x ∈ Z (j) l , namely dist(z k , x) → 0 when k → ∞, and that dist(T j (z k ), x) → 0 for k → ∞. With abuse of definition we say that such a point x is fixed. If there are points in the sequence {z k } k≥1 which are on the boundary of Z (j) l , we think of T j as its C 1+α extension on Z(j) l . We want to show that in Z(j) l there is only one fixed point x. By contradiction, suppose y is another fixed point and {w k } k a sequence converging to y and whose T j images converge to y as well. Suppose that Z(j) l is a convex set in such a way the segment with at most one fixed point, by taking the sum over the l we finally get

i , i = 1, • • • , M, then on each element Z (j) l , l = 1, • • • , |Z j |,
• (H5) Let U (j) := {f j Z (j) l , ∀l = 1, • • • , |Z j |}, and put U = ∪ ∞ j=1 U (
µ{x; dist(T j x, x)} ≤ γ N h M ε N B U m .
and this bound is independent of j.

As a consequence of Lemma 2.8 we have, Theorem 6.4 Assume a piecewise expanding map T on a compact set Z ⊂ R n satisfies conditions (H1)-(H5) and is mixing with respect to its absolutely continuous invariant measure µ. For µ a.e. p if B i (p) are nested balls about p such that µ(B i ) = 1 i . Then

a 2 n = E[( n j=1 (1 B i • T i - 1 i )) 2 ] = log n + O(1) and 1 √ log n n j=1 (1 B i • T i - 1 i ) → N(0, 1)
in distribution.

7 Discussion.

There are several natural questions remaining unanswered. In particular can the CLT for shrinking targets be proved for Anosov systems or non-uniformly hyperbolic diffeomorphisms? Chernov and Kleinbock have proved the SBC property for balls in Anosov systems [START_REF] Chernov | Dynamical Borel-Cantelli lemmas for Gibbs measures[END_REF] but the SBC property is unknown for non-uniformly hyperbolic diffeomorphisms. More generally can a limit theory be developed for the statistics of non-stationary stochastic processes arising as observations (which change in time)

on deterministic dynamical systems which may also may evolve in time, such as sequential dynamical systems?

8 Appendices 8.1 Gal-Koksma Theorem.

We recall the following result of Gal and Kuksma as formulated by W. Schmidt [START_REF] Schmidt | A metrical theory in diophantine approximation[END_REF][START_REF] Schmidt | Metrical theorems on fractional parts of sequences[END_REF] and stated by Sprindzuk [START_REF] Vladimir | Metric theory of Diophantine approximations[END_REF]: for all r = 1, . . . , log κ i.

Proof. Let C 2 = max ρ(x), C 1 = min ρ(x) where ρ(x) = dµ dm (x) is the density of µ with respect to Lebesgue measure m.

Let σ ≥ 1 and γ > σ. We choose ǫ k so that for all x a ball of radius ǫ k about x, 1 A k (y)ρ(y) dm(y).

If x ∈ F k then M k > C 1 k σ-γ .
A theorem of Hardy and Littlewood ([10] Theorem 3.17 where 0 < τ < τ . We need to alter τ to τ to take into account the fact that a ball of radius ǫ has measure roughly ǫ D . Choosing σ < γ < σ(1 + τ ) and σ > 1 the series k m(F k ) converges.

So for m a.e. x 0 there exists an N(x 0 ) such that x 0 ∈ F k for all k > N(x 0 ). Since m(B(x, ǫ k ) -m(B(x, ǫ k+1 )) ≤ 2C 2 k 2 this implies that for µ a.e. x ∈ X there exists η > 0 , κ > 0 such that for all sufficiently large i, if B i (x) is a sequence of nested balls about x, µ(B i (p)) ∼ 1 i then µ(B i (x) ∩ T -r B i (x)) ≤ µ(B i (x)) 1+η for 1 ≤ r ≤ log(i) κ . This is Assumption (C).

  and for every n ∈ N X n,i is a martingale difference array with respect to F i .We will then verify conditions(a), (b), (c) and (d) of Theorem 2.3. The hard part lies in establishing (b). This is in contrast with the stationary setting where condition (b) is usually a straightforward consequence of the ergodic theorem. Condition (b) is established in [29] by using [36, Lemma 3.3.], however in our setting the Lipschitz norms of the observations φi are unbounded and other techniques have to be used. Once we have established (a), (b), (c) and (d) it follows that lim n→∞ 1 an n-1 j=0 ψ j • T j → N(0, 1) in distribution. In the final step we show that 1 an n j=1

  j w j ) • T j → 0 in probability. Since the term 1 log n n j=1 φ 2 j • T j converges to 1 almost surely by the SBC property the proof is complete. Lemma 3.3 completes the proof of part (I) of the theorem. In order to show (II)

j 2 so

 2 it suffices to consider the sequence φ2 j + 2 wj φj . This is becausen j=1 E[ φ2 j + 2 wj φj ] = n j=1 E[φ 2 j + 2w j φ j ] + O(1) and µ almost surely, n j=1 ( φ2 j • T j + 2( wj φj ) • T j ) = n j=1 (φ 2 j • T j + 2(w j φ j ) • T j ) + O(1). Note that both wj and φj are positive functions. Let E n := n j=1 E[ φ2 j + 2 wj φj ]. We will use Proposition 8.1, a form of the Gal and Koksma theorem as stated by Sprindzuk (see Appendix) to show that 1 En n j=1

Remark 5 . 2

 52 For one-dimensional maps of the interval, Proposition 5.1 is basically a consequence of Conze and Raugi [6, Theorem 5.1]. Follow the proof of [6, Theorem 5.1] taking T k = T for all k, m to be the invariant measure µ and choosing f n =

osc( 1 A

 1 , B ǫ (p)) dLeb(p) ≤ c 1 ǫ. Then we must divide by ǫ α . As α ≤ 1 we have the ratio bounded by c 1 * (ǫ 0 ) 1-α .

  each of which is the closure of its interior, the map T j is injective and of class C 1+α on an open neighborhood of Z

Lemma 6 . 3

 63 Under the assumptions (H1)-(H5) Assumption (C) is satisfied.Proof. Denote E k (ε) := {x; dist(f k x, x) ≤ ε}.By Lemma 8.2 (see Appendix) it is enough to prove that there exists C > 0, δ > 0 such that for all k and ε, µ(E k (ε))) < Cε τ .

Proposition 8 . 1 Lemma 8 . 2

 8182 Let (Ω, B, µ) be a probability space and let f k (ω), (k = 1, 2, . . .) be a sequence of non-negative µ measurable functions and g k , h k be sequences of realnumbers such that 0 ≤ g k ≤ h k ≤ 1, (k = 1, 2, . . . , ). Suppose there exists C > 0 such that m<k≤n (f k (ω) -g k ) 2 dµ ≤ C m<k≤n h k ( * )for arbitrary integers m < n. Then for any ǫ > 01≤k≤n f k (ω) = 1≤k≤n g k + O(Θ 1/2 (n) log 3/2+ǫ Θ(n))for µ a.e. ω ∈ Ω, where Θ(n) = 1≤k≤n h k .8.2 Assumption (C) for expanding systemsIn this appendix we show that if we defineE k (ǫ) := {x : d(T k x, x) ≤ ǫ}and if the invariant measure has a density bounded above with respect to Lebesgue then assumption C is valid. Suppose µ has a density ρ with respect to Lebesgue measure which satisfies 0 < C 1 < ρ < C 2 and there exists C > 0, δ > 0 such that for all k, ǫ,µ(E k (ǫ)) < Cǫ δThen for µ a.e. p ∈ X there exists η(p) ∈ (0, 2) and κ(p) > 1 such that for all i sufficiently large µ(B i (p) ∩ T -r B i (p)) ≤ µ(B i (p)) 1+η

  denoted B(x, ǫ k ), satisfies C 1 /k σ ≤ µ(B(x, ǫ k )) ≤ C 2 /k σ . Let A k := {x : d(T j x, x) ≤ ǫ k for some 1 ≤ j ≤ log(k) ρ }. Evidently A k ⊂ log ρ k j=1 E j . By the estimate on E k (ǫ) for all large k, µ(A k ) ≤ Cǫ τ k where τ < δ. Let F k := {x : µ(B(x, ǫ k ) ∩ A k ) ≥ 1/k γ }and define the Hardy-Littlewood maximal functionM k for φ(x) = 1 A k (x)ρ(x) by M k (x) := sup a>0 1 m(B a (x)) Ba(x)

  ) states that m(|M k | > C) ≤ c 3 1 A k ρ 1 C for some constant c 3 , where • 1 is the L 1 norm with respect to m. Hence m(F k ) ≤ m(M k > C 1 k σ-γ ) ≤ µ(A k )C 1 k γ-σ ≤ k γ-σ(1+τ )

  only the norming constants change) but for simplicity we discuss in detail only the case µ(B n (p)) = 1 n . 1 Bn(p) may not lie in B α but we assume we may take an approximation to it, φα Cn k where C, k are independent of n;

	(i) |1 Bn(p) -φα n | 1 ≤ 1 n 3 and;
	(ii) φα n α ≤ (iii) φα n ≥ 0, φα n ≥ φα n+1
	Remark 2.2 If we are taking a Hölder approximation then condition (ii) is satisfied
	for the balls B i = B(p, r i ) if there exists δ(p) > 0 and C > 0 such that µ{x : r <
	d(x, p) < r + ǫ} < Cǫ δ(p) . This condition is satisfied if the invariant measure µ has
	a density h with respect to Lebesgue measure m such that h ∈ L 1+η (m) for some η > 0.
	n
	such that:

This condition could be greatly simplified as follows. Suppose the boundaries of Z i are C 1

Acknowledgements MN and LZ were partially supported by the National Science Foundation under Grant Number DMS-1101315. Part of this work was done while MN was visiting the Centre de Physique Théorique in Marseille with a French CNRS support. SV was supported by the CNRS-PEPS Mathematical Methods of Climate Theory, by the ANR-Project Perturbations and by the PICS ( Projet International de Coopération Scientifique), Propriétés statistiques des systèmes dynamiques detérministes et aléatoires, with the University of Houston, n. PICS05968. Part of this work was done while he was visiting the Centro de Modelamiento Matemático, UMI2807, in Santiago de Chile with a CNRS support (délégation).

[x, y] is contained in Z(j) l 2 . We now fix η small enough and take k big enough and such that dist(x, z k ), dist(x, T j (z k )), dist(y, w k ), dist(y, T j (w k )), are all smaller than η. We also put D m,j := inf{||DT j (x)||} > 1, where the inf is taken over the points x where the derivative is defined. The norm is the operator norm, which is strictly larger than 1 since the map is uniformly expanding. Then we have

and by applying Taylor's formula

which gives a contradiction, since D m,j > 1, by sending η to 0. Hence x is the only fixed point.

Let us now take a measurable set V ⊂ Z(j) l containing the fixed point x ∈ Z(j) l . We require that the diameter of the image T j (V ) be at most ε; such an image will therefore be contained in the ball of center T j (x) and of radius ε. The Lebesgue measure of this ball will be equal to γ N ε N , where the factor γ N was defined in the preceding footnote. Then we have

for a suitable point κ ∈ Zj l , where in the last inequality we used a local change of variable and the continuity of DT j , finally κ is a point in Z(j) l . By distortion, we could replace this point by another one, say ι such that Leb(T j (Z

l ). We therefore get (with the constant B from (BD))

Since the density of the absolutely continuous invariant measure µ is bounded from above (remember it is in L ∞ (Leb)), by, say, h M , and since each Z (j) l will contribute 2 If not we could join x and y with a chain of segments contained each in Z(j) l : the argument will work again since the sum of the lengths of those segments is larger than the distance between x and y and this is what we need in bounding from below.


W Bahsoun

H Hu

S Vaienti

Pseudo-Orbits, Stationary Measures and Metastability

We study random perturbations of multidimensional piecewise expanding maps. We characterize absolutely continuous stationary measures (acsm) of randomly perturbed dynamical systems in terms of pseudo-orbits linking the ergodic components of absolutely continuous invariant measures (acim) of the unperturbed system. We focus on those components, called least-elements, which attract pseudoorbits. We show that each least element is in a one-to-one correspondence with an ergodic acsm of the random system. Moreover our result permits to identify random perturbations that exhibit a metastable behavior.

1

Introduction

In this chapter we study statistical aspects of random perturbations of a multidimensional piecewise expanding map 1 T : M → M, M ⊂ R q . A random orbit {x ε n } n is a random process where, for all n, x ε n+1 is a random variable whose possible values are obtained in an ε-neighbourhood of T x ε n according to a transition proba- bility P ε (x n , .). We consider the case where P ε (x n , .) is absolutely continuous with respect to Lebesgue measure m. Denoting the density of the transition probability by p ε (x, .), we define a perturbed transfer operator , L ε , by:

L ε f (x) = M p ε (y, x) f (y)dy
where f ∈ L 1 m (M). We focus on non-invertible dynamical systems whose transfer operators, perturbed and non-perturbed 2 , satisfy a uniform Lasota-Yorke inequality. Among other things, such an inequality implies the existence of a finite number of ergodic acim for the initial system T , and a finite number of ergodic acsm for the random system.

We then define an equivalence relation between the ergodic components of acim of T using pseudo-orbits. Using this equivalence relation, we consequently introduce equivalence classes of ergodic acim. Among the latter, we identify those which attract pseudo-orbits and call them least elements. We show that each least element admits a neighbourhood which supports exactly one ergodic acsm of the random system, and the converse is also true, namely the support of any ergodic acsm contains only one least element. This result allows us to identify the attractors of the random orbits, namely the least elements, and we give a nice illustration in the example 2 in Section 5. Moreover, we use our result to identify random perturbations that exhibit a metastable behavior. Such a phenomenon has recently been a very active topic of research in both ergodic theory [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] Dolgopyat | The Diffusion Coefficient for Piecewise Expanding Maps of the Interval with Metastable States[END_REF][START_REF] Froyland | Lyapunov exponents, escape rates, and topological entropy in random dynamical systems[END_REF][START_REF] Gonzaléz-Tokman | Approximating invariant densities for metastable systems[END_REF][START_REF] Keller | Rare events, escape rates, quasistationarity: some exact formulae[END_REF] and applied dynamical systems [START_REF] Dellnitz | Seasonal variability of the subpolar gyres in the Southern Ocean: a numerical investigation based on transfer operators[END_REF][START_REF] Santitissadeekorn | Optimally coherent sets in geophysical flows: A new approach to delimiting the stratospheric polar vortex[END_REF].

Section 2 contains the setup of the problem, our assumptions, the notion of a least element and the statement of our main result (Theorem 1). Section 3 contains the proof of Theorem 1. In Section 4 we use the results of the previous sections to identify random systems which exhibit a metastable behavior. In Section 5 we apply our results to random transformations, in particular, we provide examples to illustrate the results of Sections 2 and 4.

2 Setup and statement of the main result

The initial system

Let M ⊂ R q be compact 3 with intM = M. We denote by d the Euclidean distance on R q . Let (M, A , m) be the measure space, where A is Borel σ -algebra, and m is the normalized Lebesgue measure on M; i.e, m(M) = 1.

Let T : M → M be a measurable map, D T be the set of discontinuities of T4 . We assume that m(D T ) = 0, and T is non-singular with respect to m. The transfer operator (Perron-Frobenius) [START_REF] Baladi | Positive transfer operators and decay of correlations[END_REF] associated with T , L :

L 1 m → L 1 m is defined by duality: for g 1 ∈ L 1 m and g 2 ∈ L ∞ m M g 1 g 2 • T dm = M L g 1 g 2 dm.

The perturbed system

We perturb the map T by introducing a family of Markov chains (X ε n ), ε > 0, n ≥ 0 with state space M and transition probabilities {P ε (x, .)} x∈M ; i.e, P ε (x, A) is the probability that a point x is mapped into a measurable set A. At time n = 0, (X ε 0 ) can have any probability distribution. We assume that: (P1) For all x ∈ M, P ε (x, .) is absolutely continuous with respect to the Lebesgue measure. We will denote its density by p ε (x, .).

Therefore P ε (X ε n+1 ∈ A|X ε n = x) = A p ε (x, y)dm(y). (P2) We have: suppP ε (x, .) = B ε (T x) for all x.
Assumption (P2) can be weakened by supposing that suppP ε (x, .) is a subset of a slightly larger ball around T x. Our proofs can be easily adapted to show that the results of this chapter will still hold under such a slightly weakened assumption.

Random orbits and stationary measures

The perturbed evolution of a state x ∈ M will be represented by a random orbit:

Definition 1. A sequence {x ε n } n≥0 ⊂ M is an ε-random orbit if each x ε n+1 is a ran- dom variable whose distribution is P ε (x ε n , .), namely {x ε n } n≥0 coincides with the Markov chains (X ε n ), ε > 0, n ≥ 0.
The counterpart of an invariant measure in the case of randomly perturbed dynamical systems is called a stationary measure:

Definition 2. A probability measures µ ε is called a stationary measure if for any A ∈ A µ ε (A) = M P ε (x, A)dµ ε (x).
We call it an absolutely continuous stationary measure (acsm), if it has a density with respect to the Lebesgue measure (see below).

The transfer operator of the random system

To study Markov processes it is useful to define the transition operator T ε acting on bounded real-valued measurable functions g defined on M:

(T ε g)(x) = M g(y)P ε (x, dy).
Its adjoint T * is defined on the space M (M) of all finite signed measures and is given by:

T * ε µ(A) = M P ε (x, A)dµ(x).
The measure µ ε is stationary if and only if

T * ε µ ε = µ ε . Moreover, a stationary mea- sure is ergodic if, any A ∈ A with T ε 1 A = 1 A implies that µ ε (A) = 0 or µ ε (A) = 1.
Since condition (P1) implies the absolute continuity of any stationary measure, it will be convenient to define an operator, L ε , acting on densities. That is to say, if µ is an absolutely continuous measure with respect to m, whose density is a function g ∈ L 1 m , then T * ε µ is an absolutely continuous measure whose density is L ε g, where L ε g is given by:

L ε g(x) = M p ε (z, x)g(z)dz. (1)
Thus, densities of acsm are fixed points of L ε . We will comment again about this definition of the random transfer operator in Remark 5, Section 5.

A Banach space and quasi-compactness of L and L ε

We now introduce a Banach space B(M) ⊂ L 1 m . We assume that (B1) The constant function 1 belongs to B(M). m norm. We also assume that the transfer operators L and L ε satisfy a uniform Lasota-Yorke inequality: there exist an η ∈ (0, 1) and a D ∈ (0, ∞) such that for all f ∈ B(M) and ε > 0 small enough:

|L f | ≤ η | f | + D f 1 ;
(LY )

|L ε f | ≤ η | f | + D f 1 . (RLY )
Assumptions (LY) and (RLY) ensure the quasi-compactness of both L and L ε , see [START_REF] Baladi | Positive transfer operators and decay of correlations[END_REF] and [START_REF] Aytac | Laws of rare events for deterministic and random dynamical systems[END_REF] for non-invertible systems and [START_REF] Demers | Stability of statistical properties in two-dimensional piecewise hyperbolic maps[END_REF] and reference therein for invertible ones in higher dimensions. In particular, among other things, (LY) implies the existence of a finite number of T -ergodic acim, and (RLY) implies the existence of a finite number of ergodic acsm for the Markov process (X ε n ). More precisely, we have for the operator L (see [START_REF] Boyarsky | Laws of Chaos, Invariant measurtes and Dynamical Systems in one dimension[END_REF][START_REF] Hennion | Limit theorems for Markov chains and Stochastic Properties of Dynamical Systems by Quasicompactness[END_REF]):

• The subspace of non-negative fixed points w of L , is a convex set with a finite number of extreme points

w 1 , • • • , w l with supports Λ k , k = 1, • • • , l. The supports Λ k , k = 1, • • • , l, are mutually disjoint Lebesgue a.e.. • The measures µ 1 = w 1 m, • • • , µ l = w l
m are ergodic and they give the ergodic decomposition of any acim µ = hm, h ∈ L 1 m . We also call them the extreme (ergodic) points (measures) decomposing µ.

The Lasota-Yorke inequality (RLY) of L ε ensures that the random system admits finitely many ergodic acsm µ ε

1 = h ε,1 m, . . . , µ ε K = h ε,K m.
Note that in general the number K of extreme points for µ ε is different from the number l of ergodic components of µ. In our setting, (see Corollary 1), we show that the number of ergodic ascm is bounded above by the number of ergodic acim.

Remark 1. We point out that for certain perturbations one can prove that (RLY) follows from (LY). See [START_REF] Baladi | On the spectra of randomly perturbed expanding maps[END_REF][START_REF] Keller | Stochastic stability in some chaotic dynamical systems[END_REF] for precise examples.

A well known consequence of assumption (LY) and (RLY) is the following proposition, see for instance [START_REF] Cowieson | Stochastic stability for piecewise expanding maps in R d[END_REF]. Proposition 1. . Let {h ε } ε>0 be a family of densities of absolutely continuous stationary measures of (X n ε ). Then any limit point, as ε → 0, of {h ε } ε>0 in the L 1 mnorm is a density of T -acim.

Pseudo-orbits, least elements and the statement of the main result

We now introduce the notion of a pseudo orbit which will be our main tool to characterize ergodic acsm. Pseudo-orbits were previously used by Ruelle [START_REF] Ruelle | Small random perturbations of dynamical systems and the definition of attractors[END_REF], followed by Kifer [START_REF] Kifer | Attractors via random perturbations[END_REF], to study attractors of randomly perturbed smooth maps.

Definition 3 (pseudo-orbit). For ε > 0, an ε-pseudo-orbit is a finite set {x i } n i=0 ⊂ M such that d(T x i , x i+1 ) < ε for i = 0 . . . n -1.
Using pseudo-orbits, we define a pre-order (reflexive and transitive) "→" among the supports {Λ i } of T -ergodic acim by writing Λ i → Λ j if for any ε > 0 there is an

ε-pseudo-orbit {x i } k i=0 such that x 0 ∈ Λ i and x k ∈ Λ j . Then we define a relation "∼" among {Λ i } by writing Λ i ∼ Λ j if both Λ i → Λ j and Λ j → Λ i .
By ergodicity, given a point y ∈ Λ i ≡ suppµ i and ε > 0, µ i -almost any point x ∈ Λ i will enter the ball B ε (y) of positive µ i measure, and therefore all the pairs (x, y) ∈ Λ i can be connected with a (finite) ε-pseudo-orbit. Hence we get an equivalence relation among those ergodic components and we define Λi the equivalence class which contains

Λ i . We write Λi → Λ j if Λ k → Λ l for any Λ k ∈ Λi and Λ l ∈ Λ j . Definition 4. Λi is said to be a least element, if there is no Λ j , j = i, such that Λi → Λ j .
This in particular means that for all ε small enough no ε-pseudo-orbit can travel from the least element to other equivalence classes. We point out that, in general, a dynamical system may have more than one least-element 5 . We now state our first result:

Theorem 1. Under Assumption (P1) and (P2), if (LY) and (RLY) hold for functions in a Banach space B(M) satisfying (B1-B3), then we have:

1. If Λ is a least element, then for ε small enough there exists an open neighborhood U ε ⊃ Λ which supports a unique ergodic acsm µ Λ ε . 2. If Λ is not a least element, then for ε small enough, µ ε ( Λ ) = 0 for any acsm µ ε .

Therefore, for any weak-limit of µ ε as ε → 0, Λ is a set of measure 0.

Theorem 1 implies the following three corollaries:

Corollary 1.
The number of acsm of the random system (X n ε ) is bounded above by the number of acim of the map T . In particular, if T has a unique acim, then (X n ε ) has a unique acsm.

We also have the converse of part 1 of Theorem 1, namely: Corollary 2. The support of any ergodic acsm contains exactly one least element.

Therefore, for ε small enough, we can uniquely associate to each least element Λ the family of densities {h Λ ε } ε>0 .

Definition 5. We say that the system (M, A , T ) is strongly stochastically stable if any L 1 m limit point of the densities of the ergodic absolutely continuous stationary measures {µ ε j } ε>0 , j = 1, . . . , K and as ε goes to zero, is a convex combination of the densities of the absolutely continuous ergodic extreme measures of µ.

In our setting, by Proposition 1 and Theorem 1, any limit point of the family {h Λ ε } ε>0 as ε goes to 0, is a convex combination of the densities of the ergodic measures spanning Λ 6 . Hence we proved that Corollary 3. The system (M, A , T ) is strongly stochastically stable. Remark 2. Our definition of stochastic stability is inspired by the definition in Sect. 1.1 of [START_REF] Alves | Random perturbations of nonuniformly expanding maps[END_REF]. Whenever there are only one absolutely continuous ergodic invariant measure µ and only one absolutely continuous ergodic stationary measure µ ε , it makes sense to speak of strongly stochastic stability of the measure µ if the density of µ converges in L 1 m to the density of µ ε , see for instance [START_REF] Cowieson | Stochastic stability for piecewise expanding maps in R d[END_REF]. Adapting this point of view we can restate the previous corollary by saiyng that: an absolutely continuous invariant measure for the original system (M, A , T ) whose support is the union of least elements, is strongly stochastically stable.

Proofs

We first prove a key lemma.

Lemma 1. Let f ∈ B ≡ B(M) and {x i } N i=0 ⊂ M be an ε-pseudo-orbit such that x j ∈ M \ (D L j ε f ∩ D T ), 0 ≤ j ≤ N and L i ε f (x i ) > 0 for some 0 ≤ i < N. Then for all i < k ≤ N we have L k ε f (x k ) > 0.
Proof. Let f ∈ B and {x i } N i=0 be an ε-pseudo-orbit, satisfying the assumptions of the lemma. In particular, suppose that for some fixed 0

≤ i < N, L i ε f (x i ) > 0. Then L i+1 ε f (x i+1 ) = M L i ε f (y)p ε (y, x i+1 ) dy.
By the hypothesis (P2) we have x i+1 ∈ B ε (Ty) ⇒ p ε (y, x i+1 ) > 0 and by the preceding continuity assumptions there exists δ > 0 such that y ∈ B δ (x i ) ⇒ L i ε f (y) > 0. But this δ -neighborhood of x i can be made smaller in such a way that when y belongs to it, d(Ty, T

x i ) ≤ ε-d(T x i ,x i+1 ) 2
which implies that x i+1 is ε-close to Ty. Therefore for all the y in this δ -neighborhood (which is of positive Lebesgue measure), the integrand above is strictly positive and this finishes the proof of the Lemma.

Proof (of Theorem 1). We first show that for every least element Λ there exists a neighborhood U ε such that T (U ε ) ⊂ U ε . We denote by B ε (A) the (open) εneighborhood of a set A, that is, B ε (A) = {x ∈ M : d(x, A) < ε}. We observe that even though a least element Λ is forward invariant, the image of a ball of radius ε centred at a point in Λ may not be necessarily contained in Λ . However, this ball will surely be a subset of the open set U ε,1 := B ε (T Λ ). We define inductively a family of nested open sets U ε,n := B ε (T U ε,n-1 ) and consider the open neighborhood U ε of Λ defined by U ε := ∪ ∞ n=1 U ε,n . This set is clearly forward invariant under T , and its closure is disjoint, for ε small enough, from the supports of the other ergodic acim; otherwise, we can construct an ε pseudo-orbit linking the least element to them.

Let

B(U ε ) := { f ∈ B| f is supported on U ε }. Since L ε f (x) = M p ε (y, x) f (
y)dy, the forward invariance of U ε together with x ∈ B ε (Ty), insure that L ε leaves the Banach space B(U ε ) invariant. Then by applying on B(U ε ) the Lasota-Yorke inequality and successively the Ionescu-Tulcea-Marinescu spectral theorem, (see for instance [START_REF] Boyarsky | Laws of Chaos, Invariant measurtes and Dynamical Systems in one dimension[END_REF][START_REF] Hennion | Limit theorems for Markov chains and Stochastic Properties of Dynamical Systems by Quasicompactness[END_REF],) we obtain a fixed point

L ε h ε = h ε ∈ B(U ε ): the measure µ ε = h ε m, U ε h ε dm = 1 is therefore stationary.
We now prove that h ε is the only fixed point of L ε in B(U ε ). Suppose there is another function h ε ∈ B(U ε )) with the same property, and let us define the function ĥ = min(h ε , h ε ). Then clearly: min

(h ε , h ε ) ≤ h ε and min(h ε , h ε ) ≤ h ε and thus L ε (min(h ε , h ε )) ≤ L ε h ε and L ε (min(h ε , h ε )) ≤ L ε h ε which implies L ε (min(h ε , h ε )) ≤ min(L ε h ε , L ε h ε ). But min(L ε h ε , L ε h ε ) = min(h ε , h ε ) =
ĥ, so L ε ĥ ≤ ĥ and therefore L ε ĥ = ĥ. Let us consider h ε -ĥ. It is a nonnegative function and it satisfies L ε (h ε -ĥ) = h ε -ĥ. By Proposition 1 and by taking ε small enough, we insure that the supports of h ε and h ε will intersect the least element in a Borel set B of positive Lebesgue measure. Starting from almost any point in this set we can attain any other point in U ε with a (finite) ε-pseudoobit 7 . Take any

x 0 ∈ B \ (D h ε -h ∩ D T ) such that (h ε -ĥ)(x 0 ) > 0. For any point x ∈ U ε \ (D h ε -h ∩ D T ) there is an ε-pseudo-orbit {x i } N i=0 ⊂ M \ (D h ε -h ∩ D T )
which starts from x 0 and lands at x = x N . Hence we can apply Lemma 1 with

f = h ε -ĥ to get L k ε (h ε -ĥ)(x k ) = (h ε -ĥ)(x k ) > 0, k = 1, • • • , N. This implies that (h ε -ĥ)(x) > 0 for all x ∈ U ε \ (D h ε -h ∩ D T ), that is h ε > h ε almost
everywhere on U ε , contradicting the fact that h ε dm = h ε dm = 1. Therefore h ε = h ε almost everywhere on U ε . We get part 1 of the theorem. Now we prove part 2. Suppose Λ0 is not a least element. We will show that h ε = 0 on Λ0 for the density h ε be the density of any acsm. We proceed by contradiction. In this case we have Λ0 → Λ for some least element Λ , if otherwise Λ0 would be a least element itself. So there is an ε-pseudo-orbit which starts from Λ0 and ends up in Λ . Let h ε denote the density of the unique acsm supported on Λ . If h ε = 0, we can invoke again the arguments of Lemma 1 and part 1 of this theorem to conclude that h ε > 0 on Λ too and also that min(h ε , h ε ) is a fixed point of L ε . But the support of such a minimum is a subset of U ε since for ε small enough, h ε = 0 outside U ε . Then by uniqueness of the density h ε over U ε we get that h ε = min(h ε , h ε ). This implies that M h ε dm > 1, which is false.

Proof (of Corollary 3). By Proposition 1 and part 2 of Theorem 1, for ε small enough, the support of any ergodic acsm µ ε 1 intersects the support of a least element. By part 1 of Theorem 1 a small neighborhood of this least element supports an ergodic acsm µ ε 2 . By repeating the arguments of the proof of part 2 of Theorem 1 we obtain that those two ergodic acsm must coincide. Finally the unicity of the least element inside the support of µ ε 1 follows from the fact that two disjoints least elements are at strictly positive distance and therefore they cannot share the same ergodic acsm.

Pseudo-orbits and metastability

An ergodic dynamical system is said to be metastable if it possesses regions in its phase space that remain close to invariant for long periods of time. A well-known approach for detecting such a behaviour is by proving that the corresponding Perron-Frobenius operator has a large real eigenvalue, ξ ε , very close to 1. Then the positive and negative parts of the eigenfunction corresponding to ξ ε can be used to identify sets which remain close to invariant for long periods of time. Such sets are often called almost invariant sets. For more information on almost invariant sets we refer the reader to [START_REF] Froyland | Escape rates and Perron-Frobenius operators: open and closed dynamical systems[END_REF] and references therein. An analogous theory also exists in the framework of non-autonomous dynamical systems, where the analogous sets are called coherent structures (see for instance [START_REF] Froyland | Lyapunov exponents, escape rates, and topological entropy in random dynamical systems[END_REF] and references therein).

In this section we assume that: (M1) As an operator on B(M), L has 1 as an eigenvalue of multiplicity two. Moreover, if λ = 1 is an eigenvalue of L , then |λ | < 1. (M2) The map T has a unique least element.

Under conditions (M1) and (M2), we will show that random perturbations of T exhibit a metastable behavior. In particular, we will show that L ε , as an operator on B, will have 1 as a simple eigenvalue and will have another real eigenvalue ξ ε close to 1. Moreover ξ ε has the second largest modulus among eigenvalues of L ε . Such a ξ ε determines the rate of mixing [START_REF] Baladi | Positive transfer operators and decay of correlations[END_REF] of the random system (X ε n ).

For this purpose, we first introduce some notation and recall the Keller-Liverani perturbation theorem [START_REF] Keller | Stability of the spectrum for transfer operators[END_REF]. We adapt it to our situation which deals with the two adapted norms

|| • || B = | • | + || • || 1 and || • || 1 .
For the unperturbed Perron-Frobenius operator L let us consider the set

V δ ,r (L ) = {z ∈ C : |z| ≤ r or dist(z, σ (L )) ≤ δ },
where σ (L ) is the spectrum of L as an operator on B. Further, for ε ≥ 0, we define the following operator norm

|||L ε ||| = sup || f || B ≤1 ||L ε f || 1 .
(

) 2
Conditions (LY) and (RLY) are necessary for the operators L , L ε to satisfy the assumptions [START_REF] Keller | Stability of the spectrum for transfer operators[END_REF]. Thus, we are ready to state and use the following important result of [START_REF] Keller | Stability of the spectrum for transfer operators[END_REF]:

Theorem 2. [18] If lim ε→0 |||L ε -L ||| = 0 then for sufficiently small ε > 0, σ (L ε ) ⊂ V δ ,r (
L ). Moreover, in each connected component of V δ ,r (L ) that does not contain 0 both σ (L ) and σ (L ε ) have the same multiplicity; i.e., the associated spectral projections have the same rank.

Using Theorem 2, we show that our random system (X n ε ) exhibits metastable behavior:

Proposition 2. Suppose that • T satisfies assumptions (M1), (M2); • lim ε→0 |||L -L ε ||| = 0.
Then, as an operator on B, L ε has 1 as a simple eigenvalue. Moreover, L ε has a real eigenvalue ξ ε very close to 1. In particular, ξ ε has the second largest modulus among eigenvalues of L ε .

Proof. Assumption (M1) states that the spectrum of L , as an operator on B, satisfies the following: ∃ an r ∈ (η, 1)8 and a δ > 0 such that:

1. The eigenvalue 1 of L is of multiplicity two; 2. if λ i = 1 is an eigenvalue of L , then λ i ∈ B(0, r);

3. B(0, r) ∩ B(1, δ ) = / 0.
Moreover, under assumptions (M2), using Theorem 1, the random map (X n ε ) has exactly one ergodic acsm; i.e., as an operator on B, L ε has 1 as a simple eigenvalue. Consequently, by Theorem 2, for sufficiently small ε, the spectrum of L ε satisfies the following:

1. L ε has a real eigenvalue ξ ε < 1, with ξ ε ∈ B(1, δ ); 2. if λ i,ε / ∈ {1, ξ ε } is an eigenvalue of L ε , then λ i,ε ∈ B(0, r).
Remark 3. The condition lim ε→0 |||L -L ε ||| = 0 of Proposition 2 can be checked in several cases. A general theorem is presented in Lemma 8 of [START_REF] Baladi | On the spectra of randomly perturbed expanding maps[END_REF] for piecewise expanding maps of the interval endowed with our pair of adapted spaces where the noise is represented by a convolution kernel. In the multidimensional case, using quasi-Hölder spaces, the proof is given in Proposition 4.3 of [START_REF] Aytac | Laws of rare events for deterministic and random dynamical systems[END_REF]. It should be noted that the previous result implies that the non-essential spectrum of L is stable [START_REF] Keller | Stability of the spectrum for transfer operators[END_REF].

Remark 4. The technique followed to prove Proposition 2 does not work when the number of ergodic T -acim is l > 2. This is due to the fact that a) If l is odd, then l -1 is even. Therefore, the transfer operator L ε may have l -1 complex eigenvalues of modulus one sitting in B(1, δ ). b) If l is even, then l -2 is even. Therefore, the transfer operator L ε may have l -2 complex eigenvalues of modulus one sitting in B(1, δ ).

Whether Proposition 2 is true or not for l > 2 is an interesting question.

Random Transformations

In sections 2, 3 and 4 we studied random perturbations of a dynamical system in the framework of general Markov processes. Nevertheless, it is often useful to deal with the case when the Markov process is generated by random transformations [START_REF] Kifer | Ergodic theory of random transformations[END_REF]. In this setting, we consider an i.i.d. stochastic process (ω k ) k∈N with values in Ω ε and with probability distribution θ ε . We associate with each ω ∈ Ω ε a map T ω : M → M and we consider the random orbit starting from the point x and generated by the realization

ω n = (ω 1 , ω 2 , • • • , ω n ), defined as : T ω n := T ω n • • • • • T ω 1 (x)
. This defines a Markov process X ε with transition function

P(x, A) = Ω ε 1 A (T ω (x))dθ ε (ω), (3)
where A ∈ B(M), x ∈ M and 1 A is the indicator function of a set A. The transition function induces an operator U * ε which acts on measures µ on (I, B(M)) as:

U * ε µ(A) = M Ω ε 1 A (T ω (x))dθ ε (ω)dµ(x) = I U ε 1 A (x)dµ ε (x),
where U ε is the random evolution operator acting on L ∞ m functions g:

U ε g = Ω ε g • T ω dθ ε (ω). (4)
We are interested in studying X ε -acsm. By (4), one can define the transfer operator L ε (Perron-Frobenius) acting on L 1 (M, B(M), m) by:

(L ε f )(x) = Ω ε L ω f (x)dθ ε (ω), (6)
which satisfies the duality condition

M gL ε f dm = M U ε g f dm ( 7
)
where g is in L ∞ m and L ω is the transfer operator associated with T ω . It is well know that µ ε := ρ ε m is a X ε -acsm if and only if L ε ρ ε = ρ ε ; i.e., ρ ε is a X εinvariant density. In order to use the results of sections 2, 3 and 4, we also assume that assumptions (P1), (P2) 9 (B1)-(B3), (LY) and (RLY) hold. Moreover, we assume that (6) reduces to

(L ε f )(x) = Ω ε L ω f (x)dθ ε (ω) = M p ε (z, x) f (z)dz. (8)
In fact, an important example of a random perturbation where L ε can be reduced as in [START_REF] Cowieson | Stochastic stability for piecewise expanding maps in R d[END_REF] is the case of additive noise. For instance if M = S q , the q-dimensional torus 10 , let define T ω = T (x)ω mod S q , where ω ∈ S q . Let the density of θ ε with respect to the Lebesgue measure dω on S m , h ε , be continuously differentiable with support contained in the square

Ω ε ≡ [-ε, ε] q : dθ ε = h ε (ω)dω = 1. It is then straightforward to check that p ε (x, y) = h ε (T x -y).
Remark 5. The preceding example illustrates very well the relation between the two approaches used in this paper to deal with randomness. Namely the Markov chain approach, which was used to prove Theorem 1, and the random transformations approach which permits to follow the orbit of a point under the concatenation of the randomly chosen maps. Consequently, the latter allows for more explicit representation of objects like the evolution operator and the transfer operator. In fact , relation ( 8) is a general fact whenever the transition function for the Markov chain is given by the integral (see ( 3)), P(x, A) = Ω ε 1 A (T ω (x))dθ ε (ω), and the noise is absolutely continuos, namely θ ε (ω ∈ Ω ε ; T ω x ∈ A) = A p ε (x, z)dz, where A is a measurable set in M. This in particular means that we can always construct a Markov chain starting with a random transformation; the converse is also true but the random maps are very often defined in an implicit way, we defer to ( [START_REF] Kifer | Ergodic theory of random transformations[END_REF]) for the construction. Finally, We stress that Theorem 1 has been proved for an absolutely continuous noise. This means that it cannot be applied, in its actual form, to random perturbations on a finite noise space, for which the probability θ ε is an atomic measure and the random transfer operator L ε applied to the function f becomes a weighted operator of the form ∑ j≥1 L ω j f p ω j , where the weights p ω j , ∑ j≥1 p ω j = 1, are associated to the values of the random variables ω j .

1-dimensional examples

To illustrate our results, we first present two simple examples of 1-dimensional maps 11 . In Example 1 we present a map that has three ergodic components and two least elements. In Example 2 we present a map which has two ergodic components and exactly one least element. Moreover, we show that when T of Example 2 is perturbed by an additive noise, it exhibits a metastable behavior.

Example 1. In this example T : [0, 10] → [0, 10]. The graph of T is shown in Fig. 1.

T is piecewise linear and Markov with respect to the partition:

[0, 1), . . . , [START_REF] Bahsoun | Escape rates formulae and metastability for randomly perturbed maps[END_REF][START_REF] Baladi | Positive transfer operators and decay of correlations[END_REF], [5, 5.5) . . . , [9.5, 10].

One can easily check that T has exactly three ergodic acim whose supports Λ 1 ,Λ T is piecewise linear and Markov with respect to the partition:

[0, 1), . . . , [START_REF] Dellnitz | Seasonal variability of the subpolar gyres in the Southern Ocean: a numerical investigation based on transfer operators[END_REF][START_REF] Demers | Stability of statistical properties in two-dimensional piecewise hyperbolic maps[END_REF].

One can easily check that that T has exactly two ergodic acim whose supports Λ 1 ,Λ 2 are, respectively, equal to [START_REF] Alves | Random perturbations of nonuniformly expanding maps[END_REF][START_REF] Baladi | Positive transfer operators and decay of correlations[END_REF], and [START_REF] Baladi | Positive transfer operators and decay of correlations[END_REF][START_REF] Dellnitz | Seasonal variability of the subpolar gyres in the Southern Ocean: a numerical investigation based on transfer operators[END_REF]. Moreover, one can easily check that T admits exactly one least elements. Namely, {Λ 1 ,Λ 2 }. When T is perturbed by an additive noise, by using Lemma 8 of [START_REF] Baladi | On the spectra of randomly perturbed expanding maps[END_REF], we obtain that lim ε→0 |||L ε -L ||| = 0. Consequently, by Proposition 2, the random system exhibits a metastable behavior.

A 2-dimensional example

We now present an example in higher dimensions, in particular the two-dimensional skew system Φ ω : [0, 1] × S → [0, 1] × S defined as (x , y ) = Φ ω (x, y) where x = T (x) + ωy y = 2y mod1, where S denotes the unit circle, and T : [0, 1] → [0, 1] is given by

• for x ∈ [0, a), T (x) = ( 1 2a -3)x + a; • for x ∈ [a, 1 4 ), T (x) = -2x + (a + 1 2 ); • for x ∈ [ 1 4 , 1 2 + a), T (x) = 2x + (a -1 2 ); • for x ∈ [ 1 2 + a, 3 4 ), T (x) = -2x + (2 + a); • for x ∈ [ 3 4 , 1 -a), T (x) = 2x + (a -1); • for x ∈ [1 -a, 1], T (x) = (3 -1 2a )x + (-5 2 + 1 2a + 2a
). The graph of T is depicted in Fig. 3 Fig. 3 The graph of the map T for a = 0.1. The least element for T is the closed interval [ 1 2 + a, 1a]. This induces the least element [ 1 2 + a, 1a] × S for Φ 0 on [0, 1] × S.

We considered a piecewise linear map to simply the exposition. However, this is not really needed to apply our results. In fact, a map with nonlinear branches can be used in the example as long as we keep uniform dilatation, bounded distortion, and a C 1+ smoothness. For each ω we have a different random map and we compose them by taking ω uniformly distributed, for instance, between (-ε, ε). In this case θ ε (ω) = 1 2ε dω. The positive parameter a can be chosen equal to 1 10 , in such a way that the image of the unit interval remains in [0, 1] when ε < a. It is very easy to check that the unperturbed map Φ 0 has two ergodic components which are, respectively, subsets of [a, 1 2 a] × S and [ 1 2 + a, 1a] × S and the latter is a least element. These ergodic components are with respect to absolutely continuous invariant measures Fig. 4 The least element on the right side x > 1 2 for the map Φ ω ; notice that ω on the top of the figure is just ε = 1/120. whose densities are the fixed points of the Perron-Frobenius operator associated to Φ 0 . The existence of such fixed points follow easily by obtaining a Lasota-Yorke inequality on a suitable function space, such as the space of functions of bounded variation [START_REF] Boyarsky | Laws of Chaos, Invariant measurtes and Dynamical Systems in one dimension[END_REF] or quasi-Hölder functions, [START_REF] Saussol | Absolutely continuous invariant measures for multidimensional maps[END_REF][START_REF] Hu | Absolutely continuous invariant measures for non-uniformly expanding maps[END_REF], which satisfy all the assumptions required in this paper. Notice that a Lasota-Yorke inequality can be obtained as well for the random Perron-Frobenius operator associated to the random system, by using the closeness of the perturbed maps Φ ω , |ω| ≤ ε for small ε (this means that the constants η and D in (LY) and (RLY) can be chosen to be the same for the unperturbed and the perturbed systems 12 ). According to our main theorem, there must be only one ergodic absolutely continuous stationary measure and this must be supported in a neighborhood of the least element. In Fig. 4 we show the limit set of several random orbits taken with ε = 1/120 and a = 0.1. All these random orbits accumulate in the right hand side of [0, 1] × S as predicted by our theory.

  (B2) The set of discontinuities, D f , of any function f ∈ B(M) has Lebesgue measure zero. (B3) There is a semi-norm |.| on B(M) such that the unit ball of B(M) is compact in L 1 m with respect to the complete norm || • || B ≡ |.| + . 1 , where . 1 denotes the L 1

Fig. 1

 1 Fig. 1 An example of a 1-dimensional map T with two least elements

Fig. 2

 2 Fig. 2 An example of a 1-dimensional map T with one least element

  the determinant of the jacobian matrix of Φ ω , which isDΦ ω = T (x) ω 0 2 , where |T | ∈ {|3 -1 2a |, 2}.The determinant does not depend on the noise and (any) norm of the matrix can be chosen uniformly bounded for ε small enough.

  Centre de Physique Théorique, CNRS, Université d'Aix-Marseille et Université de Toulon, and FRUMAM, Fédération de Recherche des Unités de Math ḿatiques de Marseille; address: CPT, Luminy Case 907, F-13288 Marseille Cedex 9 e-mail: vaienti@cpt.univ-mrs. fr

	ac.uk
	H. Hu
	Mathematics Department, Michigan State University, East Lansing, MI 48824, USA, e-mail:
	hu@math.msu.edu
	S. Vaienti
	UMR-7332

W. Bahsoun Department of Mathematical Sciences, Loughborough University e-mail: W.Bahsoun@lboro.

  2 ,Λ 3 are, respectively, equal to [1, 4], [5.5, 7.5], and [7.5, 9.5]. Moreover, one can easily check that T admits two least elements. Namely, {Λ 1 } and {Λ 2 ,Λ 3 }. Example 2. In this example T : [0, 10] → [0, 10]. The graph of T is shown in Fig. 2.

See[START_REF] Hu | Absolutely continuous invariant measures for non-uniformly expanding maps[END_REF] and[START_REF] Liverani | Multidimensional expanding maps with singularities: a pedestrian approach[END_REF] for recent results and for an exhaustive list of references on deterministic multidimensional expanding maps.

From now on with the notation D g we mean the set of discontinuities of the function g.

This will be illustrated in Example 1.

Note that within the general setting of this paper, we do not claim that the values of the weights in the convex combination that determine the limiting density can be easily identified. However, for certain perturbations of one dimensional maps, using insights form open dynamical systems, such weights can be determined. See[START_REF] Gonzaléz-Tokman | Approximating invariant densities for metastable systems[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].

The ergodicity insures this possibility for points in the same ergodic component; the equivalence relation allows to pass from one representative to the other in the least element, and finally the recursive construction of U ε allows to get the external points U ε /Λ .

By (LY) and (RLY), η is an upper bound on the essential spectral radius of L and the essential spectral radius of L ε .

(P1) and (P2) in this setting are analogous to: (P1) For all x ∈ I the measure P(x, •) defined above on the Borel subsets of I by P(x, A) = θ ε {ω ∈ Ω ε , T ω (x) ∈ A} is absolutely continuous with respect to Lebesgue, namely we have a summable density p ε (x, •) such that: P(x, A) = A p ε (x, y)dy; (P2) We have: support of P(x, •) coincides with B ε (T x), ∀x ∈ I.

If M is not the torus we assume that, for all ω ∈ Ω ε , T ω (M) ⊆ M.

In both examples the Banach space B is considered to be the space of functions of bounded variation.

For these kind of uniformly expanding maps those factors are basically related to the multiplicity of the intersection of the discontinuous lines, which is 2 in this example, and to the norm and

Acknowledgements H. Hu thanks the University of Toulon and CPT Marseille where parts of this work was conducted. WB would like to thank the London Mathematical Society for supporting his visit to CPT Marseille where parts of this research was conducted. SV thanks the French ANR Perturbations, the CNRS-PEPS "Mathematical Methods for Climate Models" and the CNRS Pics "Propriétés statistiques des systémes déterministes et aléatoires" N. 05968, for support. SV thanks Ph. Marie for useful discussions. We warmly thank Davide Faranda for having provided us with Fig. 4.