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STOCHASTIC STABILITY AT THE BOUNDARY OF EXPANDING MAPS VITOR ARAUJO AND ALI TAHZIBI Dedicated to C.Gutierrez on the occasion of his 60th birthday.A BSTRACT .We consider endomorphisms of a compact manifold which are expanding except for a finite number of points and prove the existence and uniqueness of a physical measure and its stochastical stability.We also characterize the zero-noi limit measures for a model of the intermittent map and obtain stochastic stability for some values of the parameter.The physical measures are obtained as zero-noi limits which are shown to satisfy Pesin´s Entropy Formula.C ONTENTS 1.Introduction 12.Preliminary results 63.Zero-noi limits are equilibrium measures 104.Existence i.m.and stochastic stability 155.Random perturbations of the intermittent map 16References 191.I NTRODUCTION Afte
r the long and deep developments in the last decades on the structural stability theory of dynamical systems,we know that this form of stability is too strong to be a generic property.Recently there has been some emphasis on the study of stochastic stability of dynamical systems,among other forms of stability.
On the one hand,one of the challenging problems of smooth Ergodic Theory is to prove the existence of ”nice”invariant measures called physical measures or sometimes SRB (Sinai-Ruelle-Bowen)measures.On the other hand,a natural formulation of stochastic stability of dynamical systems assumes the existence of physical measures.However,the characterization of zero-noi limit measures involved in the study of stochastic stability may provide ways to
2VITOR ARAUJO AND ALI TAHZIBI
婴粟花construct physical measures.In this work the study of zero-noi limit measures for endomor-phisms which are expanding except at afinite number of points yields a construction of physical measures and also their stochastic stability.
Let M be a compact and connected Riemannian manifold and T:=C1+α(M,M)be the space of C1+αmaps of M whereα>0.We write m for somefixed measure induced by a normalized volume form
on M that we call Lebesgue measure,dist for the Riemannian distance on M and · for the induced Riemannian norm on T M.
We recall that an invariant probability measureµfor a transformation T:M→M on a manifold M is physical if the ergodic basin
B(µ)= x∈M:1
STOCHASTIC STABILITY AT THE BOUNDARY OF EXPANDING MAPS3
T(x)= x+2αx1+αx∈[0,12,1](1.2) This map defines a C1+αmap of the unit circle S into itlf.The uniquefixed point is0and DT(0)=1.The above family of maps provides many interesting results in Ergodic Theory.If α≥1,i.e if the order of tangency at zero is high enough,then the Dirac mass at zeroδ0is the unique physical probability measure and so the Lyapunov exponent of Lebesgue almost all points vanishes[27].The situation is completely different for0<α<1:in this ca there exists a unique absolutely continuous invariant probability measureµSRB,which is therefore a physical measure and who basin has full Lebesgue measure[26].
Another point of interest is that the maps provide examples of dynamical systems with polynomial
decay of correlations.M.Holland has obtained even sub-polynomial rate of mixing modifying the intermittent maps[11].In particularµSRB is always mixing,when it exists.
1.1.Statement of the results.We consider additive noi applied to a map T of S with an indifferentfixed point at0and expanding everywhere el,as in example(1.2).Letα>0be fixed and consider T t:=T+t for|t|≤ε.ThenˆT:[−1/2,1/2]→C1+α(S,S),t→T t is a(smooth) family of C1+αmaps of S.
Letθεbe an absolutely continuous probability measure,with respect to the Lebesgue measure m on S,who support is contained in[−ε,ε](e.g.θε=(2ε)−1m|[−ε,ε],ε>0).This naturally induces a probability measure on T={T t,t∈[−1/2,1/2]}which we denote by the same symbol θε(the meaning being clear from the context).
In this tting it is well known that there always exist a stationary probability measureµεfor allε>0.Moreover this measure is ergodic and is the unique absolutely continuous stationary measure for(ˆT,θε)(e Subction2.2).
Let usfix nowα∈(0,1)and let
E={tδ0+(1−t)µSRB:0≤t≤1}
be the t of linear convex combinations of the Dirac mass at0with the unique absolutely con-tinuous invariant probability measure for the maps.
Theorem A.Letµ0be any accumulation point of the stationary measures(µε)ε>0whenε→0 for the random perturbation(ˆT,θε)ε>0withα∈(0,1).Thenµ0∈E.
In the caα≥1there does not exist any absolutely continuous invariant probability measure. However the Dirac measureδ0is the unique physical measure.In this ca we are able to obtain stochastic stability.
Theorem B.Letα≥1in(1.2)and let(µε)ε>0be the family of stationary measures for the random perturbation(ˆT,θε)ε>0.Thenµε→δ0whenε→0in the weak∗topology. However,taking a different family f t unfolding the saddle-node
f t(x)= tx+2α(2−t)x1+αx∈[0,12,1](1.3)
4VITOR ARAUJO AND ALI TAHZIBI
with α∈(0,1)we obtain an example of non-stochastic stability .In fact,since f ′t (0)=t then for t <1the fixed point 0is a sink for f t (e Figure 1)and we prove that the physical measure for the random sys
tem is always δ0for restricted choices of the probability measures θε.
Theorem C.For every small enough ε>0there are a (ε)<b (ε)<1such that a (ε)→1when ε→0and,for any given probability measure θεsupported in [a (ε),b (ε)],the unique stationary measure µεfor the random system equals δ0.
Since f 1=T admits an absolutely continuous invariant measure µSRB and clearly δ0cannot converge to this physical measure,we have an example of a stochastically unstable system (under this kind of perturbations).1
1111/2
y=x 1/2y=x
p y=f(x)t y=f (x)00t
F IGURE 1.The map f =T (left)and the map f t for 0<t <1(right).
Using the same kind of additive perturbations considered in Theorems A and B,our methods provide the following results for maps in higher dimensions.
Theorem D.Let f :M →M be a C 1+αlocal diffeomorphism such that
(1) D f (x )−1 ≤1for all x ∈M;
(2)K ={x ∈M : D f (x )−1 =1}is finite and |det D f (x )|>1for every x ∈K.
平房效果图Then,for any non-degenerate random perturbation (ˆf ,θε)ε>0,there exists a unique ergodic sta-tionary probability measure µεfor all ε>0.Moreover µεconverges,in the weak ∗topology when ε→0,to a unique absolutely continuous f -invariant probability measure µ0who basin has full Lebesgue measure,and f is stochastically stable.
Here we will assume that M is a n -dimensional torus since the maps f satisfying the conditions on Theorem D are at the boundary of expanding maps,which can only exist on special mani-folds [25,10],the best known example being the tori.Since the manifolds are parallelizable,we can define additive perturbations just as we did on the circle.If T n is a n -dimensional torus,then T T n ≃R n and ˆf :B ⊂R n →C 1+α(M ,M ),v →f +v ,where B is a ball around the origin of R n (together with a family (θε)ε>0of absolutely continuous probability measures on B ,e
STOCHASTIC STABILITY AT THE BOUNDARY OF EXPANDING MAPS5 Subction2.3for the defin
ition of non-degenerate random perturbation)will be a the kind of additive perturbation we will consider.
The results will be derived from the following more technical one,but also interesting in itlf.
Theorem E.Let f:M→M be a C1+αlocal diffeomorphism such that
(1) D f(x)−1 ≤1for all x∈M;滑铁卢之战
(2)K={x∈M: D f(x)−1 =1}isfinite.
Then,for any non-degenerate random perturbation(ˆf,θε)ε>0,every weak∗accumulation point µof the quence(µε)ε>0,whenε→0,is an equilibrium state for the potential−log|det D f(x)|, i.e.
hµ(f)= log|det D f(x)|dµ(x).(1.4) Moreover every equilibrium stateµas above is a convex linear combination of an absolutely continuous invariant probability measure withfinitely many Dirac measures concentrated on periodic orbits who Jacobian equals1.
Cowieson and Young have prented results similar to ours for C2or C∞diffeomorphisms. However their assumptions are on the convergence of the sum of the positive Lyapunov exponents for the ran
dom maps to the same sum for the original map,and they obtain SRB measures,not necessarily physical ones,e[9]for more details.We make much stronger assumptions on both the kind of maps being perturbed(expanding except atfinitely many points)and the kind of perturbations ud(additive besides being absolutely continuous),and we obtain physical measures for C1+αendomorphisms.
In what follows,wefirst prent some examples of applications and then general results about random dynamical systems(Section2)to be ud to prove Theorem E(Section3).At this point we are ready to obtain Theorem D(Section4).Finally we apply the ideas to the specific ca of the intermittent maps(Section5),completing the proof of Theorems A,B and C.
1.2.Examples.In what follows we write T for S×S and consider S=[0,1]/{0∼1}.We always assume that the spaces are endowed with the metrics induced by the standard Euclidean metric through the identifications.The Lebesgue measure on the spaces will be denoted by m (area)on T and m1(length)on S.
An extra example is the intermittent map itlf,dealt with in Section5.
1.2.1.Direct product“intermittent×expanding”.Let f:T→T,(x,y)→(Tα(x),g(y)),where Tαis defined at the Introduction withα>0,and g:S→S is C1+α,admits afixed point g(0)=0 and g′=Dg>1.
Since f is a direct product,ifα∈(0,1),then f admits an invariant probability measureν=µα×λ,whereµαis the unique absolutely continuous invariant measure for Tαandλis the unique absolutely continuous invariant measure for ,µα≪m1andλ≪m1.Hence the product measure is absolutely continuous:ν≪m=m1×m1.The measures are ergodic and also mixing,and the basins ofµαandλequal S,m1mod0.Thus their direct productνis ergodic and so B(ν)=T,m mod0.
6VITOR ARAUJO AND ALI TAHZIBI
Ifα≥1,thenν=δ0×λis again an ergodic invariant probability measure for f with B(ν)= T,m mod0,sinceλis the same as before and so is mixing for g,andδ0is Tα-ergodic,with the basin of both measures equal to S.
Here K={0}×S(the definition of K is given at the statement of Theorem D)is notfinite, and the conclusion of Theorem D does not hold whenα≥1:we have a physical measure which is not absolutely continuous with respect to m.Note that clearly (D f)−1 ≤1everywhere and since K containsfixed(and periodic)points,f is not uniformly expanding.
1.2.2.Direct product“intermittent×intermittent”.Let f:T→T,(x,y)→(Tα(x),Tβ(y))where α,β>0.Now K={0}×S∪S×{0}and,by the same reasoning of the previous example,the probability measureν=µα×µβis the
unique physical measure for f.Moreover B(ν)=T as before.Howeverνis absolutely continuous with respect to m if,and only if,α,β∈(0,1).
1.2.3.Skew-product“intermittent⋊expanding”.Let f:T→T,(x,y)→(Tα(x)+ηy,g(y)),for α>0,η∈(0,1),and g:S→S as in example1.2.1.
In this ca we easily calculate D f= DTαη0Dg and so K={(0,0)}.
Clearly (D f)−1 ≤1everywhere and since K is afixed point the map f is not uniformly expanding.Applying Theorem D we get an absolutely continuous invariant probability measure µfor f with log (D f)−1 dµ<0.Hence the Lyapunov exponents for Lebesgue almost every point on the basin ofµare all positive,so f is a non-uniformly expanding transformation.
This map is stochastically stable,since every weak∗accumulation point of(µε)ε>0whenε→0 equalsµby the uniqueness part of Theorem D.We stress that since the value ofαplayed no role in the arguments,the conclusions hold for anyα>0.
2.P RELIMINARY RESULTS
Throughout this ction we outline some general results about random dynamical systems to be u
d in what follows.
Having a parameterized family of mapsˆT:X→T,t→T t,where X is some connected com-pact metric space,enables us to identify a quence T1,T2,...of maps from T with a quence ω1,ω2,...of parameters in X.The probability measureθεcan then be assumed to be supported on X.
We tΩ=X N,the space of quencesω=(ωi)i≥1with elements in X.Then we endowΩwith the standard infinite product topology,which makesΩa compact metrizable space,with distance given by(for example)d(ω,ω′)=∑j≥12−1d X(ωj,ω′j)where d X is the distance on X.We also take the standard product probability measureθε=θNε,which makes(Ω,B,θε)a probability space.Here B=B(X)is theσ-algebra generated by cylinder ts,that is,the minimal σ−algebra of subts ofΩcontaining all ts of the form{ω∈Ω:ω1∈A1,ω2∈A2,···,ωl∈A l} for any quence of Borel subts A i⊂X,i=1,···,l and l≥1.
