Extreme Value Theory for Piecewise Contracting Maps with Randomly Applied Stochastic Perturbations

0.28 MB·English

by  D. Faranda

Extreme Value Theory for Piecewise Contracting Maps with
Randomly Applied Stochastic Perturbations
Davide Faranda ∗ Jorge Milhazes Freitas † Pierre Guiraud ‡ Sandro Vaienti §
5
1 March 19, 2015
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2

r
a
Abstract
M
Weconsidergloballyinvertibleandpiecewisecontractingmapsinhigherdimensionsandweperturb

7 themwithaparticularkindofnoiseintroducedbyLasotaandMackey. Wegotrandomtransformations
1 which are given by a stationary process: in this framework we develop an extreme value theory for a
few classes of observables and we show how to get the (usual) limiting distributions together with an

] extremal indexdependingon thestrength of the noise.
S
D
h. 1 Introduction
t
a
m
Inapreviousnote[7]weannouncedafewresultsontheapplicationofExtremeValueTheory(EVT)toone-
[ dimensionalpiecewisecontractingmaps(PCM)wheneverthelatterareperturbedwithaparticularkindof

noise. ThisnoisewasintroducedinthebookbyLasotaandMackey[16]anditwascalledrandomly applied
2
v stochastic perturbation (RASP). It turns out that it is particularlyuseful to study PCM especially because
3 it allows very efficient representations of the transfer operator (Perron-Frobenius) and of the stationary
1 measures. We now generalize to higher dimensional PCM our previous results by proving the existence of
9
extreme value laws (EVL) for some kinds of observables associated to rare visits in the neighborhood of a
2
given point. As a matter of fact, we first consider globally invertible maps defined on compact subsets of
0
Rn andwhicharediffeomorphismsbutonsomesingular(Borel)setsofzeroLebesguemeasure,sotheyare
.
1
not necessarilycontractions. The RASP belongs to the class of random transformations (see next section),
0
which naturally define a Markov chain with associated a unique absolutely continuous stationary measure
5
1 (see Th. 10.4.2 in [16]). By taking the direct product of this stationary measure with the probability
: distributionoftheconcatenatedmaps,wegetthe(annealed)probabilityPwhichwillgovernthestatistical
v
propertiesofoursystem. Inparticularthestochasticprocessgivenbytheobservationalongarandomorbit
i
X will be stationary according to P. We will first prove a general result about decay of correlations which
r will not require strong assumptions on the functional spaces of the observables (Proposition 2.1). We will
a
therefore use this result to recover some sort of asymptotic independence in order to prove convergence of
the distribution of the maxima toward the Gumbel’s law. This will be accomplished in two other steps:
first we will get an explicit expression of the affine coefficients defining the sequence of levels u needed to
n
rescale the distribution of the maximum to avoiddegenerate limits (Propositions 4.1,4.2,4.3); then we will
control the short returns in Proposition 4.5. Whenever we will consider rare visits in the neighborhood of
aperiodic cycle,wewillprovethe existenceofanextremalindexlessthan 1,andinparticularequaltothe
magnitude of the noise.
We would like to stress that we already investigated the EVT for random transformations obtained by
perturbing with additive noise; we applied it to uniformly expanding maps [1] and even to rotations [8].
∗Laboratoire SPHYNX, Service de Physique de l’Etat Condensé, DSM, CEA Saclay, CNRS URA 2464, 91191 Gif-sur-
Yvette, France. e-mail: <[email protected]>.
†Centro de Matemática & Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto,
Portugal. e-mail:<[email protected]>.
‡CIMFAV,FacultaddeIngeniería, UniversidaddeValparaíso,Valparaíso,Chile. e-mail: <[email protected]>.
§AixMarseilleUniversité,CNRS,CPT,UMR7332,13288Marseille,FranceandUniversitédeToulon,CNRS,CPT,UMR
7332,83957LaGarde, France. e-mail:<[email protected]>.
1
In both cases we got the existence of an EVL around any point and with extremal index equal to 1. It
seems that RASP is much more suitable for piecewise contractions and does not smoothen out completely
the periodicity featuresofthe unperturbedmap, asthis is reflectedinthe persistency ofanextremalindex
less than one. A few numerical simulations seem to suggest however the existence of an EVL even when
we perturb PCM with additive noise [7].
WenowbrieflyrecallthemainstatementsoftheEVT.Letussupposethat(Yn)n∈N isasequenceofreal-
valuedrandomvariablesdefinedontheprobabilityspace(Ψ,P).Wewillbeinterestedinthedistributionof
themaximumM :=max{Y ,Y ,...,Y }whenn→∞.Itiswellknownthatthe limitingdistributionis
n 0 1 n−1
degenerateunless one proceedto a suitable re-scalingof the levelsof exceedances. The precise formulation
is the following: we have an Extreme Value Law for (Mn)n∈N if there is a non-degenerate distribution
function H : R → [0,1] with H(0) = 0 and, for every τ > 0, there exists a sequence of levels (un(τ))n∈N
such that
lim nP(Y >u )→τ, (1)
0 n
n→∞
and for which the following holds:
lim P(M ≤u )→1−H(τ).
n n
n→∞
The motivation for using a normalizing sequence (un)n∈N satisfying (1) comes from the case when
(Yn)n∈N are independent and identically distributed. In this i.i.d. setting, it is clear that P(Mn 6 u) =
(F(u))n, being F(u) the cumulative distribution function for the variable u. Hence, condition (1) implies
that
τ n
P(M 6u )=(1−P(Y >u ))n ∼ 1− →e−τ,
n n 0 n
n
(cid:16) (cid:17)
as n → ∞. Note that in this case H(τ) = 1−e−τ is the standard exponential distribution function. Let
us now choose the sequence u = u (y) as the one parameter family u = y/a +b , where y ∈ R and
n n n n n
a > 0, for all n ∈ N. Whenever the variables Y are i.i.d. and for some constants a > 0, b ∈ R,
n i n n
we have P(a (M −b ) 6 y) → G(y), where the convergence occurs at continuity points of G, and G is
n n n
non-degenerate, then G will converge to one of the three EVLs: Gumbel, Fréchet or Weibull. The law
n
obtained depends on the the common distribution of the random variables.
When Y ,Y ,Y ,... are not independent, the standard exponential law still applies under some con-
0 1 2
ditions on the dependence structure. These conditions will be stated in detail later and they are usually
designatedwithD′ andD2;whentheyholdfor(Yn)n∈N thenthereexistsanextremevaluelawforMn and
H(τ) = 1−e−τ, see Theorem 1 in [11]. We want to stress that these two conditions alone do not imply
the existence of an extreme value law; they require, even to be checked, that the limit (1) holds. It turns
outthatforthe kindofobservableswearegoingtointroduce,andwhicharerelatedto the localproperties
of the invariant measure, the limit (1) is difficult to prove when the invariant measure is not absolutely
continuous,since one needs the exactasymptotic behavior of that measure on small balls. Instead it turns
out that whenever the systems is randomly perturbed and the invariant measure is absolutely continuous,
the existence of the limit (1) can be insured with general arguments and in the majority of the cases (see
below for a precise meaning), it canbe expressedby a closedformula in terms of the strength of the noise,
Proposition 4.2 and 4.3 below. Moreoverthat formula could be used in a reversedway: since the sequence
(un)n∈N is now uniquely determined for any n, a numerical sampling for (un)n∈N which provides conver-
genceto the extremevalue law,will bringinformationonthe localgeometricalpropertiesofthe stationary
measure: this approachwas successfully used, for instance, in [9, 10].
2 Random dynamical systems: RASP perturbation
Let us consider a sequence of i.i.d. random variables (Wk)k∈N with values (ωk)k∈N in a space Ωε and with
commonprobabilitydistributionθ . LetX ⊂RD beacompactsetequippedwiththeLebesguemeasurem
ε
definedontheBorelσ-algebra,and(f ) afamilyofmeasurabletransformationssuchthatf :X →X
ω ω∈Ωε ω
2
for all ω ∈ Ω 1. Given a point x ∈ X and a realization ω = (ω ,ω ,...) ∈ ΩN of the stochastic process
ε 1 2 ε
(Wk)k∈N, we define the random orbit of x as the sequence (fωn(x))n∈N, where
f0(x)=x and fn(x)=f ◦f ◦···◦f (x) ∀n>1.
ω ω ωn ωn−1 ω1
The transformations f will be considered as stochastic perturbations of a deterministic map f, in the
ω
sense that they willbe takenin a suitable neighborhoodoff whosesize willbe determined by the value of
ε, see below. We could therefore define a Markov process on X with transition function
L (x,A)= 1 (f (x))dθ (ω), (2)
ε A ω ε
ZΩε
where A ∈ X is a measurable set, x ∈ X and 1 is the indicator function of the set A. A probability
A
measures µ is called stationary if for any measurable set A we have:
ε
µ (A)= L (x,A)dµ (x).
ε ε ε
ZX
We call it an absolutely continuous stationary measure, if it has a density with respect to the Lebesgue
measure.
We are now ready to introduce the randomly applied stochastic perturbations. We first have a Borel
measurable map f acting on the compact set X; then we operate an aleatory reset of the state of the
system at each failure of a Bernoulli random variable: if (xn)n∈N denotes the successive states of such a
random dynamical systems, then at each time n ∈ N we have x = f(x ) with probability (1−ǫ) and
n+1 n
x =ξ with probability ǫ, where ξ is the realization of a random variable with value in X. This kind
n+1 n n
of perturbation corresponds to the family (f ) of random transformations defined by
ω ω∈Ωε
f (x)=ηf(x)+(1−η)ξ ∀x∈X, (3)
ω
where ω =(η,ξ) is a random vector with value in Ω ={0,1}×X. The two components η and ξ of ω are
ε
independent and η is a Bernoulli variable with the probability of being 0 equal to ε, while ξ is a random
variablethatwe willsupposeLebesgue-uniformlydistributedonX2. The jointdistributionθ ofthese two
ε
componentsis the productofthe Bernoullimeasurewith weights(ε,1−ε)andthe uniformmeasureon X.
In order to obtain the stationary measure µ , let us introduce the random Koopman operator U :
ε ε
L∞ →L∞ defined for all φ∈L∞3 by
U φ(x):= φ(f (x))dθ .
ε ω ε
Z
Now, if we take two observables φ∈L∞ and ψ ∈L1, it easy to check that
U φ(x)ψ(x)dx = φ(f (x))ψ(x)dθ dx=
ε ω ε
Z Z Z
ε φ(x)ψ(y)dxdy+(1−ε) φ(f(x))ψ(x)dx = φ(x)P ψ(x)dx,
ε
Z Z Z Z
where P is the adjoint operator of U , that is the random transfer operator. If we denote P the transfer
ε ε
operator associated to f and ψ = ψ(y)dy, then we have
R
P ψ(x)=(1−ε)Pψ(x)+εψ. (4)
ε
1Inthefollowingwhenwewillrefertoadynamicalsystem(X,f,µ)wewillmeanthatf isdefinedonX andpreservesthe
Borelprobabilitymeasureµ;ifwewillwrite(X,f),thiswillsimplycorrespond totheactionoff onX.
2InthebookofLasotaMackay[16]thevariableξismoregenerallydistributedwithadensitywithrespecttotheLebesgue
measure;thiswillnotchangetheresults ofourpaper.
3From now on L1 andL∞ willbereferred to theLebesgue measure mandthe integral withrespect to thelatter willbe
denoteasR(·)dx.
3
The stationary measure µ verifies φ(x)dµ = U φ(x)dµ and in our case is given by µ = h m where
ε ε ε ε ε ε
h ∈L1 is a density such that h =P h . Such a density exists and is given by [16]:
ε ε ε ε
R R
∞
h =ε (1−ε)kPk1. (5)
ε
k=0
X
In the random setting we are developing the decay of correlations will be of annealed type and it will
be formulated in terms of the iterates of the random Koopman operator, in particular we have:
Proposition 2.1. If φ∈L∞ and ψ ∈L1∩L∞, then
φ(fωn(x))ψ(x)dµεdθεN− φ(x)dµε ψ(x)dµε 62(1−ε)n||φ||L∞||ψhε||L1. (6)
(cid:12)Z Z Z Z (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Proof. To us(cid:12)e the duality of the Koopman operator with the t(cid:12)ransfer operator P, let us introduce the
following quantity:
Cor (φ,ψ,n):= Un(φ(x))ψ(x)dx− φ(x)dµ ψ(x)dx = φ(x)Pnψ(x)dx− φ(x)dµ ψ(x)dx .
m ε ε ε ε
(cid:12)Z Z Z (cid:12) (cid:12)Z Z Z (cid:12)
(cid:12) (cid:12) (cid:12) (7)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
By recurrence, it(cid:12)is easy to obtain the following formula for t(cid:12)he i(cid:12)terates of P : (cid:12)
ε
n−1
Pnψ =(1−ε)nPnψ +εψ (1−ε)kPk1. (8)
ε
k=0
X
By replacing this expression into (7) and writing there dµ =h dx, with h given by (5), we get
ε ε ε
∞
Cor (φ,ψ,n) = (1−ε)n φ(x)Pnψ(x)dx−εψ φ(x) (1−ε)kPk1(x)dx
m
(cid:12) (cid:12)
(cid:12) Z Z kX=n (cid:12)
(cid:12) ∞ (cid:12)
(cid:12) (cid:12)
6 (cid:12)(1−ε)n||φ||L∞||Pnψ||L1 +ε|ψ|||φ||L∞|| (1−ε)kPk1||L1 (cid:12)
k=n
X
6 2(1−ε)n||φ||L∞||ψ||L1,
by Hölder inequality and the fact that the infinite sum is of positive terms. Therefore, if ψ ∈ L1∩L∞,
then the correlations (6) are equal to
Corm(φ,ψhε,n)62(1−ε)n||φ||L∞||ψhε||L1.
Remark 2.2. We will argue later on that in the EVT the preceding decay of correlations result plays a
fundamental role whenever ψ is a specific characteristic function: in this case it will be sufficient to have
the density h in L1. For observables ψ which are solely in L1 we will need a density in L∞. Note that
ε
the exponential decay of the correlations does not depend on the map f, which is a remarkable property of
RASP.
Wearenowreadytointroducetheextremevaluestatistics;atthisregardwedefineastochasticprocess
(Yn)n∈N, n∈N, by composing a given observable φ:X →R as: Yn =φ◦fωn, with
f (x)=η f(x)+(1−η )ξ , ∀x∈X,
ωn n n n
where ω = (η ,ξ ) ∈ {0,1}×X is random variable which distribution is the product of the Bernoulli
n n n
measure with weights (ε,1−ε) and the uniform measure on X.
In this case (Yn)n∈N will be a stationary process if we consider the probability P = µε ×θεN, which
corresponds to the annealed situation where we average over the initial condition and over the realization
of the noise.
4
We will in particular choose the observable φ(x) = −logd(x,z), where d(·,·) denotes (some) distance
onX andz apointofX. AsweanticipatedintheIntroduction,suchanobservableisrelatedtorecurrence
in small sets, since the distribution of the maximum of the random observable φ◦fk,k = 0,...,n−1 up
ω
to the level u , coincides with the distribution of the first entrance of the random orbit into the ball (in
n
the metric d), B(z,e−un) centered at z and of radius e−un
We now state the two conditions which insure weak dependence of the process and which allow to get
the limiting distribution of the maxima.
Condition[D (u )] We say that D (u ) holds for the sequence Y ,Y ,... if for all ℓ,t and n,
2 n 2 n 0 1
|P(Y >u ∩max{Y ,...,Y 6u })−P(Y >u )P(M 6u )|6γ(n,t),
0 n t t+ℓ−1 n 0 n ℓ n
where γ(n,t) is decreasing in t for each n, and nγ(n,t )→0 when n→∞ for some sequence t =o(n).
n n
Now, let (kn)n∈N be a sequence of integers such that
k →∞ and k t =o(n). (9)
n n n
Condition[D′(u )] We say that D′(u ) holds for the sequence Y ,Y ,Y ,... if there exists a sequence
n n 0 1 2
(kn)n∈N satisfying (9) and such that
⌊n/kn⌋
lim n P(Y >u ,Y >u )=0. (10)
0 n j n
n→∞
j=1
X
As we said in the Introduction when these two conditions hold for (Yn)n∈N then there exists an extreme
value law for M and H(τ)=1−e−τ, but provided that the limit (1) is true.
n
3 Stationary measure for injective piecewise diffeomorphisms with
RASP
We now introduce the dynamical system upon which we will perform the randomly applied stochastic
perturbation.
Definition 3.1. The map. Let us X be a compact subset of RD which is the closure of its interior
X =int(X), equipped with the metric d and with the normalized Lebesgue measurem; suppose {X }N is a
i i=1
collection of N disjoint open subsets of X such that X = N X and m(∆)=0, where ∆:=X\ N X
i=1 i i=1 i
is the singularset. We will consider non singular maps f :X →X which are injective and such that f|
X\∆
is a C1-diffeomorphism. We say that f is continuous in aSpoint w of the boundary of X if there isSan open
ball of radius κ, B(w,κ) such that f is continuous on B(w,κ)∩int(X) and it can be extended continuously
on B(w,κ).
Remark 3.2. In the examples we will treat, the set ∆ will be composed by points where f is discontinuous
and by the points of the boundary of X. We would like to stress that a point of that boundary is not
necessarily a discontinuity point of f.
The following proposition give the expression of the stationary density.
Proposition 3.3. Let {Λk}k∈N be the sequence of sets defined by Λ0 :=X and Λk+1 :=f(Λk\∆) for all
k >1. Denote J (x):= k |det(f′(f−l(x))|−1 for all k >1 and J (x):=1(x). Then,
k l=1 0
Q ∞
h (x)=ε (1−ε)kJ (x)1 (x) ∀x∈X. (11)
ε k Λk
k=0
X
5
Note that J (x) is welldefined for everyx∈X suchthat x∈/ fl(∆) for alll∈{1,...,k}. Among other
k
basicspropertiesofthe sets Λ whichwillbe helpful allalongthis paper,the followinglemma ensuresthat
k
J (x) (and therefore h (x)) is well defined for all x∈Λ .
k ε k
Lemma 3.4. The sequence of sets {Λk}k∈N has the following properties:
1. Λ is open for all k >1
k
2. Λ ⊂Λ for all k ∈N,
k+1 k
3. Λ ∩fp(∆)=∅, for all p∈{1,...,k}.
k
Proof. 1) The set Λ is open since X \∆ is open and f is a diffeomorphism in X \∆. Using the same
1
arguments,weobtainbyinductionthatΛ isopenforanyk >1. 2)WehaveΛ =f(Λ \∆)⊂f(X\∆)=
k 2 1
Λ and if we suppose Λ ⊂Λ , then Λ =f(Λ \∆)⊂ f(Λ \∆) =Λ . 3) By injectivity we have
1 k k−1 k+1 k k−1 k
f(X \∆)∩ f(∆) = ∅, i.e. Λ ∩f(∆) = ∅. Suppose now that Λ ∩ fk(∆) = ∅ and that there exists
1 k
x ∈ Λ ∩fk+1(∆). Then there exist y ∈ Λ \∆ and y′ ∈ fk(∆) such that f(y) = f(y′) = x. As f is
k+1 k
injective,wemusthavey =y′,whichcontradictsthe recurrencehypothesis. ItfollowsthatΛ ∩fk(∆)=∅
k
for all k >1 and applying 1) we obtain the desired result.
Proof. (Proposition 3.3). We compute the transfer operator P of f in order to use the formula (5). Since
f is a diffeomorphism on the full Lebesgue measure open set X \∆, its associated transfer operator acts
on a function ψ ∈L1(X) as
Pψ(x)=1 (x)ψ(f−1(x))|det(f′(f−1(x))|−1 =1 (x)ψ(f−1(x))J (x) ∀x∈X.
f(X\∆) Λ1 1
Since the setf(∆)haszeroLebesguemeasure,wecansupposewithoutlossofgeneralitythatPψ(x)=0if
x∈f(∆). Then, Pψ(x) is well defined for all x∈X, since f(X\∆)=Λ does not contain point of f(∆).
1
Now, we show by induction that for the uniform density 1=1 we have
X
Pk1(x)=1 (x)J (x) ∀k >1.
Λk k
It is easy to check that P1(x)=1 (x)J (x) and that
Λ1 1
Pk+11(x)=1 (x)1 (x)J (x),
Λ1 f(Λk) k+1
if we suppose the property true at rank k. Now, we have
Λ ∩f(Λ )=(Λ ∩Λ )∪(Λ ∩f(Λ ∩∆))=Λ ∪(Λ ∩f(Λ ∩∆)).
1 k 1 k+1 1 k k+1 1 k
Since Λ ∩ f(Λ ∩ ∆) ⊂ Λ ∩ f(∆) = ∅, we have Λ ∩ f(Λ ) = Λ . It follows that Pk+11(x) =
1 k 1 1 k k+1
1 (x)J (x), which ends the induction. Using (5) we obtain that
Λk+1 k+1
∞
h (x)=ε (1−ε)kJ (x)1 (x) ∀x∈X,
ε k Λk
k=0
X
which ends the proof of the proposition.
Now we decompose the space X as the disjoint union X = Λ ∪f(∆)∪(X \f(X)) and we use (11)
1
to study the density in each of these sets. Since the last two sets do not belong to any Λ with k 6= 0,
k
we obtain that h = ε on these sets. For x ∈ Λ , by property 2) of Lemma 3.4, two situations can occur:
ε 1
either there exists p>1 such that x∈Λ \Λ or x∈Λ:= ∞ Λ . So in Λ we have on one hand
p p+1 k=1 k 1
p T
h (x)=ε (1−ε)kJ (x) ∀x∈Λ \Λ , (12)
ε k p p+1
k=0
X
and the by continuity of the J , the density is continuous. On the other hand we have
k
∞
h (x)=ε (1−ε)kJ (x) ∀x∈Λ. (13)
ε k
k=0
X
6
In this last case the density may not be bounded, depending on the behavior of J (x). However, if we
k
suppose that there exists 0< λ< (1−ε)−1 such that |det(f′(x)|−1 6λ for all x ∈X \∆, then the series
(13) convergesuniformly inΛ andh is alsocontinuousin Λ andthus onthe wholeset Λ (see Proposition
ε 1
4.3).
In the next sections we decompose our study of the existence of EVL in two parts. The first part
concerns the point of Λ \Λ˜, where
1
Λ˜ := Λ . (14)
k
k∈N
\
The set Λ˜ is called the global attractor of f and contains all the point of Λ. In Λ \Λ˜ we can verify the
1
conditions D (u ) and D′(u ) to show the existence of EVL, but in Λ˜ condition D′(u ) is not verified.
2 n n n
For this reason in the second part, we introduce other conditions which allow us to show the existence of
EVL but with an extremal index different from 1; this is achieved by giving additional property, namely
by choosing f as a piecewise contracting map. It has been shown in [4] that for any piecewise contracting
map defined in a compact space, if the global attractor Λ˜ does not intersect the set of the discontinuities,
then it is composed of a finite number of periodic orbits. Moreover, this condition, for injective maps as
our, is generic in C0 topology [3]. All these properties will help to get the existence of EVL in the set Λ˜.
Let us give a few example to illustrate that
Example 3.1. As an easy example of an uncountable hyperbolic attractor, let us consider the baker’s map
f defined on the unit square [0,1]×[0,1] iteratively for n≥1, by
γ x , for y <α
x = a n n
n+1 1 +γ x , for y >α
(cid:26) 2 b n n
1y , for y <α
y = α n n
n+1 1 (y −α)+γ x , for y >α
(cid:26) 1−α n b n n
with 0 ≤ x ,y ≤ 1, 0 < γ < γ < 1, α ≤ 1. In this case ∆ is given by the union of the boundary of the
n n a b 2 2
square and the segment y =α. Moreover Λ is given by the two rectangles [0,γ ]×[0,1], [1,1+γ ]×[0,1].
1 a 2 2 b
Notice that the derivative f′ is a diagonal matrix with diagonal elements γ (or γ ), and 1 (or 1 ). We
a b α 1−α
can choose these parameters in order to get det(f′)<1.
TypicalexamplesofPCMforwhichtheattractorisgenericallyafinitesetofperiodicorbitsarepiecewise
affine[2,17]. Thatis,ifforeachi∈{1,...,N}therestrictionf :=f| ofthemapf :X →X ofDefinition
i Xi
3.1 to a piece X has the following form:
i
f (x)=A x+c ∀x∈X ,
i i i i
where A :RD →RD is a linear contraction and c ∈X.
i i
Example 3.2. 1-D case The simplest example of a piecewise contracting map of this class is given in
dimension 1 in the interval [0,1] by f(x) = ax+ c with a,c ∈ (0,1) (here X = (0,(1−c)/a), X =
1 2
((1−c)/a,1) and ∆ = {0,1,(1−c)/a}). For almost all the values of the parameters a and c, the global
attractor Λ is composed of a unique periodic orbit (which period depends on the specific values of the
parameters), but for the remaining set of parameters the global attractor is a Cantor set supporting a
minimal deynamics. We refer to [5] for a detailed description of the asymptotic dynamics of this map. We
remind that in our paper [7] we investigated in detail the case with c=0 and 0 the (unique) fixed point; we
briefly quote here the simple structure of the density
p−1(1−ε)k
h (x)=ε ∀x∈(ap,ap−1], p≥1
ε ak
k=0
X
and the formulae for the coefficients of the linear scaling. At this regard let τ > 0, y = −ln(τ) and
u = y +b with
n an n
p−1(1−ε)k
a =1 and b =log 2nε ∀n∈N,
n n αk
!
k=0
X
7
If:
(i) z 6=0 on the interval but not in the countably many discontinuity points of hε, namely z ∈/ ∪j∈N{aj};
(ii) p>1 such that z ∈(ap,αp−1) and n is large enough such that the ball B(z,e−un)⊂(ap,ap−1), then:
nP(X >u )=τ
0 n
Example 3.3. 2-Dcase IntheunitsquareX =[0,1]2, anotherinterestingexampleis given byamapwith
four pieces X :=(0,T )×(0,T ), X :=(T ,1)×(0,T ), X :=(0,T )×(T ,1) and X :=(T ,1)×(T ,1),
1 1 2 2 1 2 3 1 2 4 1 2
singular set ∆ given by the two segments {x = T },{y = T } and the boundary of the unit square, and
1 2
restricted maps f (x,y) := a(x,y)+(1−a)(1,0), f (x,y) := a(x,y)+(1−a)(1,1), f (x,y) := a(x,y)
1 2 3
and f (x,y) := a(x,y)+(1−a)(0,1). Once again, for almost all values of the parameter a ∈ [0,1), T
4 1
and T , the global attractor is composed of a finite number of periodic orbits. However, contrarily to the
2
1 dimensional case, where the number of periodic orbits is bounded by the number of pieces N [18], the
number of periodic orbits increases with the parameter a and tends to infinity when a goes to 1. This
example belongs to a larger class of higher dimension piecewise affine contracting maps which are models
for genetic regulatory networks and are studied in [6].
QuestionAn interesting question is the check if the stationary measure is stochastically stable, in the
sensethatthemeasureµ willconvergeweaklytothe SRBmeasuresupportedonthe globalattractor. Let
ε
us remind that it is not the case for the simple contraction f(x)=ax, x∈[0,1], a<1, as it was provedin
Lasota Mackay [16].
4 Convergence to the extreme value law
We prove in this section the convergence towards an extreme value distribution in the complement of the
attractor defined in (14). We will work with the process Y (x,ω) := −logd(fn(x),z). Our first task will
n ω
be to prove the existence of the limit (1); this will be accomplished in two manners: we will first give a
generalresultshowingthe existence ofthat limit forany pointofX, butwithout anexplicitexpressionfor
the scaling coefficients a and b . Such coefficients could instead be computed on the set Λ \Λ˜ and even
n n 1
onthe attractoritself providedthe density ofthe stationarymeasurewillbe essentiallybounded. The first
generalresultuses Theorem1.7.13 in the book by Leadbetter, Lindgren and Rootzen[15]: accordingto it,
a sufficient condition to guarantee the existence of the limit (1) for 0 < τ < ∞, is that 1−F(x) → 1, as
1−F(x−)
x → x , where F is the distribution function of Y , the term F(x−) in the denominator denotes the left
F 0
limit of F at x and finally x := sup{x;F(x) < 1}. For our particular observable Y , x = ∞ and the
F 0 F
probabilityPreducestothestationarymeasureµ whichisabsolutelycontinuousandthereforenonatomic.
ε
Therefore F is continuous and the above ratio goes to 1. We have thus proved the following proposition:
Proposition4.1. Supposef :X →X satisfiesDefinition3.1. Then,foranyz ∈X definingtheobservable
Y (·)=−logd(·,z) and any 0<τ <∞, there exists a sequence u such that
0 n
lim nP(X >u )=τ.
0 n
n→∞
As we anticipated above the previous result could be strengthened whenever z lies in the complement
of the global attractor under the general assumption that h is summable. Actually, we will suppose that
ε
z ∈ (Λ \Λ˜)\∪∞ ∂Λ , where ∂Λ := Λ \Λ ; this in turn implies that z ∈ Λ \Λ ⊂ Λ \Λ for
0 k>2 k k k k p p+1 p p+1
some p > 1.4 We can take advantage of the continuity of the density in such points to obtain a possible
scaling sequence for which the limit (1) holds.
Proposition 4.2. Suppose f : X → X satisfies Definition 3.1. Let p > 1 such that z ∈ Λ \Λ . Let
p p+1
τ >0, y =−ln(τ) and u = y +b with a =D and b such that
n an n n n
p −1
1
m(B(z,e−bn))= nε (1−ε)kJ (z) = ∀n∈N,
k
nh (z)
! ε
k=0
X
4Wenoticethatinthecaseofthebaker’stransformationofExample3.1theboundariesoftheΛk belongtotheattractor
andtherefore wecantakesimplyz∈(Λ0/Λ˜).
8
where B(z,r) denotes a ball of center z and radius r. Then,
lim nP(X >u )=τ. (15)
0 n
n→∞
Proof. For all n∈N, we have
nP(X0 >un)=n 1B(z,e−un)(x)dµǫ =n 1B(z,e−un)(x)hε(x)dx.
Z Z
Since(un)n∈N isanincreasingsequenceforanyτ >0andz isintheopensetΛp\Λp+1,ifnislargeenough
B(z,e−un)⊂Λp\Λp+1. Using formula (12) and m(B(z,e−un))=τm(B(z,e−bn)) we obtain:
p p
nτm(B(z,e−bn))ε (1−ε)k inf J (x)6nP(X >u )6nτm(B(z,e−bn))ε (1−ε)k sup J (x).
k 0 n k
k=0 x∈B(z,e−un) k=0 x∈B(z,e−un)
X X
(16)
Since Jk is continuous in B(z,e−un) for all k∈{0,...,p}, we have
lim inf J (x)=J (z)= lim sup J (x).
k k k
n→∞x∈B(z,e−un) n→∞x∈B(z,e−un)
Using the expression of the sequence (un)n∈N, we obtain
lim nP(X >u )=τ.
0 n
n→∞
We note that an inspection of the proof of Proposition 4.2 easily shows that similar results can be
obtainedforthepointsofΛ,butsupposingacontinuousboundeddensity. Thefollowingpropositionshows
that the density has such properties in Λ under some condition on the derivative of f.
Proposition 4.3. Let 0<λ<(1−ε)−1 and suppose that |det(f′(x)|−1 ≤λ, for all x∈X/∆, then h is
ε
continuous in Λ and the same conclusions of Proposition 4.2 hold.
Proof. For any k ≥ 0 and x ∈ X let g (x) := ε(1−ε)kJ (x)1 (x). Then, g (x) = 0 if x 6= Λ and
k k Λk k k
g (x) ≤ ε((1−ε)λ)k if x ∈ Λ and therefore p g (x) converges uniformly on X when p → ∞, since
k k k=0 k
∞ ((1−ε)λ)k converges. Now, let x ∈Λ, then
k=0 0 P
P ∞ ∞ ∞ ∞
lim h (x)= lim g (x)= lim g (x)=ε (1−ε)k lim J (x)=ε (1−ε)kJ (x )=h (x )
ε k k k k 0 ε 0
x→x0 x→x0 x→x0 x→x0
k=0 k=0 k=0 k=0
X X X X
We have used that x ∈ Λ for all k ≥ 0, which implies on one hand by openness of the Λ that
0 k k
lim 1 (x) = 1 and in the other hand that J is continuous in x , since for any x ∈ Λ we have
x→x0 Λk k 0 k
fl(x)∈/ f(∆) for all l≤k and f is C1 in X\∆.
We are now ready to check conditions D (u ) and D′(u ) which will guarantee the convergence of our
2 n n
process towardan extreme value law, the Gumbel’s one in our case. As we said above condition D (u ) is
2 n
insured by the decay of correlations result established in Proposition 2.1, see the proof of Th. D in Sect.
4.1 in [1] for the very general approach. We are thus left with the proof of D′(u ). We first need:
n
Lemma 4.4. Let n ∈ N and Un = B(z,e−un) be a ball centered at a point z ∈ X with radius e−un. Let
p>1 be such that z ∈Λ \Λ then,
p p+1
min{j−1,p}
P((x,ω ,...,ω ):x∈U ,fj(x)∈U )=ǫµ (U ) (1−ǫ)k J (x)1 (x)dx ∀j >1, (17)
1 j n ω n ǫ n k Un
k=0 Z
X
for n large enough. In particular, under the same hypothesis we have,
P((x,ω ,...,ω ):x∈U ,fj(x)∈U )6µ (U )2 ∀j >1. (18)
1 j n ω n ǫ n
9
Proof. Let us denote
Pr :=P((x,ω ,...,ω ):x∈U ,fj(x)∈U ) ∀j >1.
j 1 j n ω n
First of all, for all j >1 we can write
Pr = 1 (x)1 (fj(x))dθNdµ = 1 (x)Uj(1 )(x)dµ = Pj(1 h )(x)1 (x)dx.
j Un Un ω ǫ ǫ Un ǫ Un ǫ ǫ Un ǫ Un
Z Z Z Z
Using (8), we obtain that for all j >1
Pr = Pj(1 h )(x)1 (x)dx
j ǫ Un ǫ Un
Z
j−1
= (1−ǫ)j Pj(1 h )(x)1 (x)dx+ǫµ (U ) (1−ǫ)k Pk1(x)1 (x)dx
Un ǫ Un ǫ n Un
Z k=0 Z
X
j−1
= (1−ǫ)j 1 (x)1 (fj(x))h (x)dx+ǫµ (U ) (1−ǫ)k J (x)1 (x)1 (x)dx
Un Un ǫ ǫ n k Λk Un
Z k=0 Z
X
j−1
= (1−ǫ)jµ (U ∩f−j(U ))+ǫµ (U ) (1−ǫ)k J (x)1 (x)dx.
ǫ n n ǫ n k Λk∩Un
k=0 Z
X
Now suppose n large enough for U being a subset of Λ \ Λ . Then one can show that fj(U ) ⊂
n p p+1 n
j fk(∆)∪Λ (using recurrence on j). Now, in the union j fk(∆) we have that Λ ∩fk(∆)= ∅
k=1 p+j k=1 p
for any k 6 p. We also have fl(Λ )∩fp+l(∆) = ∅ for any l ∈N, since Λ ∩fp(∆) = ∅ and f is injective.
p p
STogetherwith Λ ⊂Λ , this implies that U ∩fj(U )=∅, andSthereforeU ∩f−j(U )=∅ for allj >1.
p+j p n n n n
On the other hand, U ∩Λ =∅ for all k >p. It follows that
n k
min{j−1,p}
Pr =ǫµ (U ) (1−ǫ)k J (x)1 (x)dx ∀j >1,
j ǫ n k Un
k=0 Z
X
which prove (17). Now, it easy to show using (12) that Pr 6Pr =µ (U )2 for any j >1.
j p+1 ǫ n
Proposition4.5. Letp>1andz ∈Λ \Λ ,thenthemaximaoftheprocessX (x,ω):=−lnd(fn(x),z)
p p+1 n ω
verifies condition D′(u ).
n
Proof. Suppose z ∈ Λp\Λp+1 for some p > 1. Consider the sequence (un(y))n∈N of Proposition 4.2. Let
(kn)n∈N be such that kn →∞ when n→∞ and kn =o(n). Supposing n large enough and using (18), we
deduce that
⌊ n ⌋
kn n (nµ (U ))2
n P((x,ω ,...,ω ):x∈U ,fj(x)∈U )6n µ (U )2 6 ǫ n .
1 j n ω n k ǫ n k
j=1 (cid:22) n(cid:23) n
X
Using limnµ (U )=τ, we obtain that
ε n
[ n ]
kn
lim n P((x,ω ,...,ω ):x∈U ,f ◦···◦f (x)∈U )=0,
n→∞ 1 j n ωj ω1 n
j=1
X
since nµǫ(Un) → τ and kn → ∞ when n → ∞. In such a case, (Xn(x,ω))n∈N verifies conditions D2(un)
and D′(un) for our sequence (un(y))n∈N, wich implies that P(Mn 6un(y)) converges towards the Gumbel
law e−e−y.
Collecting all the results of this section we finally proved the following theorem
Theorem 4.6. Suppose f : X → X satisfies Definition 3.1. Then, for any z ∈ (Λ \Λ˜)\∪∞ ∂Λ , the
1 k>2 k
sequence {Mn}n∈N of the maxima of the process defined for every n ∈ N by Yn(x,ω) := −logd(fωn(x),z)
admits the Gumbel’s law as extreme values distribution.
10