Random Dynamical Systems

Home , Dynamical system, Stable manifold theorem

V´ıtor Araujo´ Centro de Matematica´ da Universidade do Porto Rua do Campo Alegre, 687, 4169-007 Porto, Portugal E-mail: [email protected]

1 Introduction randomness be embedded within the model to begin with. We present the most basic classes of models in what The concept of random dynamical system is a com- follows, then deﬁne the general concept and present paratively recent development combining ideas and some developments and examples of applications. methods from the well developed areas of probability theory and dynamical systems. Let us consider a mathematical model of some physi-

¥ 2 Dynamics with noise

¡ ¡ k

k ¢£¢£¢ cal process given by the iterates T0 T0 T0 ¤ k

1, of a smooth transformation T : M ¦ of a man- 0 To model random perturbations of a transformation ifold into itself. A realization of the process with §

T0 we may consider a transition from the image T0 x ¨

initial condition x0 is modelled by the sequence §

§ k to some point according to a given probability law, © T0 x0 ¨£¨ k 1, the orbit of x0. obtaining a Markov Chain, or, if T0 depends on a pa- Due to our inaccurate knowledge of the particular rameter p, we may choose p at random at each iter- physical system or due to computational or theo- ation, which also can be seen as a Markov Chain but retical limitations (lack of sufficient computational whose transitions are strongly correlated. power, inefficient algorithms or insufficiently developed mathematical or physical theory, for example), the mathematical models never correspond exactly to 2.1 Random noise

the phenomenon they are meant to model. Moreover

¦ when considering practical systems we cannot avoid Given T0 : M and a family p x ¨ : x M

either external noise or measurement or inaccuracy of probability measures on M such that the sup-

§ §

¢ ¨

errors, so every realistic mathematical model should port of p x ¨ is close to T0 x , the random or-

allow for small errors along orbits not to disturb too bits are sequences xk k © 1 where each xk 1 is a ran-

§ ¢ much the long term behavior. To be able to cope dom variable with law p xk ¨ . This is a Markov

with unavoidable uncertainty about the “correct” pa- Chain with state space M and transition probabil-

rameter values, observed initial states and even the ities p x ¨ x M . To extend the concept of in- speciﬁc mathematical formulation involved, we let variant measure of a transformation to this setting,

we say that a probability measure µ is stationary if M M, let us deﬁne

§ § §

¨ ¨

µ A ¨ p A x dµ x for every measurable (Borel)

¨ ¨£¨

m A B T x ¤ ε

ε 0

§ § §

subset A. This can be conveniently translated by say- p A x ¨

¨ ¨£¨

m B T0 x ¤ ε

ing that the skew-product measure µ p on M M

given by where m denotes some choice of Riemannian volume §

ε ¢%

¨ §

§ form on M. Then p x is the normalized volume

¨ ¨

¤£££¤ ¤£££ d µ p x0 ¤ x1 xn

restricted to the ε-neighborhood of T0 x ¨ . This de-

§ § §

¢£¢£¢ ¢£¢£¢

¨ ¨

dµ x0 ¨ p dx1 x0 p dxn 1 xn ﬁnes a family of transition probabilities allowing the §

points to “jump” from T0 x ¨ to any point in the ε-

¦ is invariant by the shift map S : M M on the neighborhood of T0 x ¨ following a uniform distribu- space of orbits. Hence we may use the Ergodic The- tion law.

orem and get that time averages of every continuous

n 1 §

§ 1

ϕ˜ x ¨ lim ϕ x ¤£££&¤ k Alternatively we may choose maps T1 ¤ T2 Tk in- n ∞ n ∑ k 0

dependently at random near T0 according to a proba- §

n 1

§ §

1 §

k bility law ν on the space T M ¨ of maps, whose sup-

lim ϕ π0 S x ¨£¨£¨ n ∞ n ∑ port is close to T in some topology, and consider 0

k 0 §

¡ ¡

¢£¢£¢

sequences xk Tk T1 x0 ¨ obtained through ran-

exist for µ p -almost all sequences x, where π0 : dom iteration, k 1 ¤ x0 M.

M M M is the natural projection on the ﬁrst This is again a Markov Chain whose transition prob-

coordinate. It is well known that stationary measures §

abilities are given for any x M by ¢!

always exist if the transition probabilities p x ¨ de-

§ § §

)(

pend continuously on x. '£

¨ ¨ ¨

p A x ν T T M : T x A ¤

A function ϕ : M is invariant if ϕ x ¨ §

§ so this model may be reduced to the ﬁrst one. How-

¨ ϕ z ¨ p dz x for µ-almost every x. We then say ever in the random maps setting we may associate that µ is ergodic if every invariant function is con- to each random orbit a sequence of maps which are stant µ-almost everywhere. Using the Ergodic Theo- iterated, enabling us to use robust properties of the rem again, if µ is ergodic, then ϕ˜ " ϕdµ, µ-almost transformation T0 (i.e. properties which are known everywhere. to hold for T0 and for every nearby map T) to derive Stationary measures are the building blocks for more properties of the random orbits.

sophisticated analysis involving e.g. asymptotic so-

Under some regularity conditions on the map x * §

journ times, Lyapunov exponents, decay of correla-

p A x ¨ for every Borel subset A, it is possible to tions, entropy and/or dimensions, exit/entrance times represent random noise by random maps on suit- from/to subsets of M, to name just a few frequent ably chosen spaces of transformations. In fact the notions of dynamical and probabilistic/statistical na- transition probability measures obtained in the ran-

ture. dom maps setting exhibit strong spatial correlation:

§ §

¢+ ¢%

# ¨ Example 1 (Random jumps). Given ε 0 and T0 : p x ¨ is close to p y is x is near y.

2 If we have a parameterized family T : U M M case suggests the name additive random perturba-

of maps we can specify the law ν by giving a proba- tions for random perturbations deﬁned using families ¤£££ bility θ on U. Then to every sequence T1 ¤£££&¤ Tk of maps of this type. of maps of the given family we associate a sequence

For the probability measure on U we may take ¤£££ ω1 ¤£££&¤ ωk of parameters in U since

θε any probability measure supported in the ε-

¡ ¡ ¡ ¡

¢£¢£¢ ¢£¢£¢ k

Tk T1 T T T neighborhood of e and absolutely continuous with

- - -.,

ωk ω1 ω1 , ωk

§ §

¥ respect to the Riemannian metric on M, for any ε 0

¨ ¨ for all k 1, where we write Tω x T ω ¤ x . In small enough. this setting the shift map S becomes a skew-product Example 4 (Local additive perturbations). If M transformation d

and U0 is a bounded open subset of M

§ § §

*/' ( ¦

¨ ¨ ¨

¤ ¤

S : M U x ¤ ω Tω1 x σ ω strictly invariant under a diffeomorphism T0, i.e.,

§ §

closure T U ¨£¨65 U , then we can deﬁne an isomet- to which many of the standard methods of dynami- 0 0 0 ric random perturbation setting

cal systems and ergodic theory can be applied, yield-

§ §

ing stronger results that can be interpreted in random ¨

V T0 U0 ¨ (so that closure V § terms. §

closure T0 U0 ¨£¨5 U0); Example 2 (Parametrical noise). Let T : P M

7 d d G 89 the group of translations of ;

M be a smooth map where P¤ M are ﬁnite dimen-

sional Riemannian manifolds. We ﬁx p P, de- 7 0 V a small enough neighborhood of 0 in G.

note by m some choice of Riemannian volume form

§ §

¨ ¨ ¤

on P, set Tw x T w x and for every ε 0 write

§ § § § §

¨ 3

Then for v and x V we set T x x v, with

1 ¢ V v

¨£¨ ¨£¨ ¤ θ m B p ¤ ε m B p ε , the normalized ε 0 0 the standard notation for vector addition, and clearly restriction of m to the ε-neighborhood of p0. Then

§ Tv is an isometry. For θε we may take any probability ¨ T together with θ deﬁnes a random perturba- w w P ε measure on the ε-neighborhood of 0, supported in V tion of T , for every small enough ε # 0. p0 and absolutely continuous with respect to the volume

Example 3 (Global additive perturbations). Let M d # in , for every small enough ε 0. be a homogeneous space, i.e., a compact connected Lie Group admitting an invariant Riemannian metric.

Fixing a neighborhood U of the identity e M we 2.3 Random perturbations of ﬂows

§ § §

¨ ¨£¨ ¤

can deﬁne a map T : U M M ¤ u x Lu T0 x ,

¢ In the continuous time case the basic model to

where Lu x ¨ u x is the left translation associated

to u M. The invariance of the metric means that start with is an ordinary differential equation dXt

§ § §

3 ¨ ¨ ¨ ¨ ¤

left (an also right) translations are isometries, hence f t ¤ Xt dt, where f : 0 ∞ X M and X M is

M ¨ the family of vector ﬁelds in . We embed random-

ﬁxing u U and taking any x ¤ v TM we get

§ 0 § 0 0 § § §

0 ness in the differential equation basically through

¢ ¢

¨£¨ ¨ ¨

DTu x ¨ v DLu T0 x DT0 x v diffusion, the perturbation is given by white noise or

0 § 0

¢ ¨

DT0 x v Brownian motion “added” to the ordinary solution.

21 d n

In the particular case of M , the d-dimensional In this setting, assuming for simplicity that M ,

§ §

¨43 torus, we have Tu x ¨ T0 x u and this simplest the random orbits are solutions of stochastic differ-

ential equations 2.4 The abstract framework

§ §

; ;

¨ 3 ¨

¤ ¤ ¤ ¤ dXt f t ¤ Xt dt ε σ t Xt dWt 0 t T X0 Z The illustrative particular cases presented can all be

written in skew-product form as follows. #

where Z is a random variable, ε ¤ T 0 and both f : §

n n n § k n

: <=> ? : <=>

¤FE

¤ ¤

0 ¤ T and σ : 0 T L are Let Ω be a given probability space, which will §

k n 1

¨ measurable functions. We have written L ¤ be the model for the noise, and be time, which

k n

G G

@ for the space of linear maps and Wt for the usually means 6¤ (discrete, resp. invertible sys-

white noise process on . The solution of this equa- tem) or H¤ (continuous, resp. invertible system).

tion is a stochastic process A random dynamical system is a skew-product

§ §

§ § § § § §

A *

¦ *

¨ ¨

¤ ¤ ¤

¨ ¨ ¨ ¨ ¨£¨

¤ ¤ ¤ ¤

X : Ω M t ω Xt ω St : Ω M ¤ ω x θ t ω ϕ t ω x

1 1

for some (abstract) probability space Ω, given by for all t , where θ : Ω Ω is a family of

§ §

¨ ¨

T T measure preserving maps θ t : Ω ¤FE and ϕ :

§ §

3CB ¨ 3CB ¨

¤ ¤ ¤ ¨ Xt Z f s Xs ds ε σ s Xs dWs Ω M M is a family of maps ϕ t ¤ ω : M

0 0 1

satisfying the cocycle property: for s ¤ t , ω Ω

where the last term is a stochastic integral in the

¨ ¤

sense of Ito.ˆ Under reasonable conditions on f and ϕ 0 ¤ ω IdM

§ § § § § ¡

with continuous

¨ ¨ ¨£¨ ¨ , there exists a unique solution 3

¤ ¤ σ ϕ t s ¤ ω ϕ t θ s ω ϕ s ω

paths, i.e.

: *

3 ¨D ¨ 0 ¤ ∞ t X ω t In this general setting an invariant measure for the

is continuous for almost all ω Ω (in general these random dynamical system is any probability measure

paths are nowhere differentiable). µ on Ω M which is St -invariant for all t and

§ § §

¨£¨ ¨ E

whose marginal is , i.e. µ St U µ U and

§ §

Setting Z the probability measure concentrated § δ

x0 1

¨£¨ ¨ 5 µ πΩ U E U for every measurable U Ω

on the point x0, the initial point of the path is x0 with

§ M, respectively, with πΩ : Ω M Ω the natural

probability 1. We write Xt ω ¨ x0 for paths of this §

§ projection.

* ¦ ¨ type. Hence x Xt ω ¨ x deﬁnes a map Xt ω : M which can be shown to be a homeomorphism and Example 5. In the setting of the previous examples of

even a diffeomorphisms under suitable conditions on random perturbations of maps, the product measure

f and σ. These maps satisfy a cocycle property η E µ on Ω M, with Ω U , θε and µ any §

§ stationary measure, is clearly invariant. However not

¨ ¨

X0 ω IdM identity map of M ¤ all invariant measures are product measures of this

§ § § § §

¨ ¨ ¨£¨ ¨ ¤

Xt s ω Xt θ s ω Xs ω type.

for s ¤ t 0 and ω Ω, for a family of measure pre- Naturally an invariant measure is ergodic if every St -

§ §

¨ ¨

serving transformations θ s : Ω ¤FE on a suitably invariant function is µ-almost everywhere constant.

I ¨

chosen probability space Ω ¤FE . This enables us to i.e. if ψ : Ω M satisﬁes ψ St ψ µ-almost 1 write the solution of this kind of equations also as a everywhere for every t , then ψ is µ-almost ev- skew-product. erywhere constant.

3 Applications the set of elements where the sequence λi jumps ex-

; actly at the indexes in I. Then for ω ΩI, 1 J h l

We avoid presenting well established applications of

§ 0 § § 1 0

P ;

¨ ¨ ¨ both probability or stochastic differential equations ΣI , h ω y : lim log ϕn ω λih ω n ∞ n

(solution of boundary value problems, optimal stop- N

ping, stochastic control etc) and dynamical systems is a vector subspace with dimension ih 1 1. §

(all sort of models of physical, economic or bio-

(2) Setting ΣI , k 1 ω 0 , then

logical phenomena, solutions of differential equa-

§ 0 § 1 0

tions, control systems etc), focusing instead on top-

¨ ¨ lim log ϕn ω λih ω ¤

ics where the subject sheds new light on these areas. n ∞ n

§ §

¨+R ¨

, for every y ΣI , h ω ΣI h 1 ω .

3.1 Products of random matrices and the

O (3) For all ω Ω O there exists the matrix

Multiplicative Ergodic Theorem

§ §

§ 1 2n

' ()T

¨ ¨VU

A ω ¨ lim ϕn ω ϕn ω S The following celebrated result on products of ran- n ∞ dom matrices has far-reaching applications on dy-

λi

namical systems theory. whose eigenvalues form the set e : i 1 ¤£££&¤ k .

§ The values of λi are the random Lyapunov charac- ¨ Let Xn n © 0 be a sequence of independent and iden- teristics and the corresponding subspaces are analo-

tically distributed random variables on the probabil- §

§ § ¨

k k gous to random eigenspaces. If the sequence Xn n © 0

¨ ¨ ¤

ity space Ω ¤ P with values in L such that

§ 0 0

is ergodic, then the Lyapunov characteristics become

¨KJL3

E log X1 ∞, where log x max 0 ¤ log x

0 0 §

¢ k k non-random constants, but the Lyapunov subspaces

and is a given norm on L ¤ . Writing

§ § §

¡ ¡

¢£¢£¢ are still random.

¨ ¨ ϕn ω ¨ Xn ω X1 ω for all n 1 and ω Ω we obtain a cocycle. If we set We can easily deduce the Multiplicative Ergodic

Theorem for measure preserving differentiable maps

§ 0 § 0

1 §

¨ ¨

B ω ¤ y Ω : lim log ϕ ω y ¨

n T0 ¤ µ on manifolds M from this result. We as-

n ∞ n §

sume for simplicity that M 5 and set p A x

§ §

exists and is ﬁnite or is ∞ ¤ and

¨ ¨

δ Y A 1 if T x A and 0 otherwise. Then the X

k T0 x 0

O P

¨ ¤

Ω ω Ω : ω ¤ y B for all y

measure µ p on M M is σ-invariant (as deﬁned

¡ ¡

O O

then Ω O contains a subset Ω of full probability and in Section 2) and we have that π0 σ T0 π0, where

there exist random variables (which might take the π0 : M M is the projection on the ﬁrst coordinate,

§ §

¥ ¥ ¥

¢£¢£¢

¨ ¨

value N ∞) λ1 λ2 λk with the following and also π0 µ p µ. Then setting for n 1 T

properties. § k k

¡ ¡

X : M L n

¢£¢£¢ and Xn X π0 σ

# # #

(1) Let I k 1 i i i 1 be any * 1 2 l 1 x DT x ¨

§ 0 ¨

l 3 1 -tuple of integers and then we deﬁne §

§ we obtain a stationary sequence to which we can ap-

O O

¨ ¨

¤ ¤

ΩI ω Ω : λi ω λ j ω ¤ ih i j ih 1 and ply the previous result, obtaining the existence of

§ §

¨ J J ¨ h l λih ω λih Q 1 ω for all 1 Lyapunov exponents and of Lyapunov subspaces on

5 a full measure subset for any C1 measure preserving in dynamical systems for decades extending to the dynamical system. present day. The random multiplicative ergodic theorem sets the stage for the study of the stability of By a standard extension of the previous setup we ob- Lyapunov exponents under random perturbations. tain a random version of the multiplicative ergodic

theorem. We take a family of skew-product maps ¦ St : Ω M as in Subsection 2.4 with an invariant

§ 3.2 Stochastic stability of physical measures

¦ ¨ probability measure µ and such that ϕ t ¤ ω : M is (for simplicity) a local diffeomorphism. We then The development of the theory of dynamical sys- consider the stationary family

tems has shown that models involving expressions

Xt : Ω L TM 1 as simple as quadratic polynomials (as the logistic

¤ t

* ¦ ¨

ω Dϕ t ¤ ω : TM family or Henon´ attractor), or autonomous ordinary §

§ differential equations with a hyperbolic singularity

¨ ¨ ¤ where Dϕ t ¤ ω is the tangent map to ϕ t ω . This is

of saddle-type, as the Lorenz ﬂow, exhibit sensitive

¤ a cocycle since for all t ¤ s ω Ω we have

dependence on initial conditions, a common feature

§ § § § ¡

of chaotic dynamics: small initial differences are

3 ¨ ¨ ¨ ¨

¤ ¤ ¤ X s t ω X s θ t ω X t ω rapidly augmented as time passes, causing two tra-

If we assume that jectories originally coming from practically indistin-

0 § § 0 §

1 guishable points to behave in a completely different

' (

¨ ¨ ¨

¤FE ¤

sup sup log Dϕ t ¤ ω x L Ω

0 Z t 1 x M manner after a short while. Long term predictions 0 0 based on such models are unfeasible since it is not where ¢ denotes the norm on the corresponding possible to both specify initial conditions with arbi- space of linear maps given by the induced norm trary accuracy and numerically calculate with arbi- (from the Riemannian metric) on the appropriate tan- trary precision.

gent spaces, then we obtain a sequence of random

¥ ¥

variables (which might take the value N ∞) λ1 λ2 ¥ ¢£¢£¢ λ , with k being the dimension of M, such that

k Physical measures

§ 0 §

1 0

¨ ¨ ¤ lim log Xt ω ¤ x y λi ω x

t ∞ t Inspired by an analogous situation of unpre-

§ §

dictability faced in the ﬁeld of Statistical Mechan-

¨ ¨%R ¨

¤ ¤

for every y Eiω ¤ x Σi ω x Σi 1 ω x and i §

§ ics/Thermodynamics, researchers focused on the

3 ¨£¨ ¤ 1 ¤£££¤ k 1 where Σi ω x i is a sequence of vector statistics of the data provided by the time averages of

subspaces in TxM as before, measurable with respect §

§ some observable (a continuous function on the man-

¨ ¨ ¤ to ω ¤ x . In this setting the subspaces Ei ω x and ifold) of the system. Time averages are guaranteed

the Lyapunov exponents are invariant, i.e. for all t §

1 to exist for a positive volume subset of initial states

¨ and µ almost every ω ¤ x Ω M we have

(also called an observable subset) on the mathemat-

§ § § § § §

¨£¨ ¨ ¨£¨ ¨

¤ ¤ ¤ λi St ω ¤ x λi ω x and Ei St ω x Ei ω x ical model if the transformation, or the ﬂow associated to the ordinary differential equation, admits a The dependence of Lyapunov exponents on the map smooth invariant measure (a density) or a physical T0 has been a fruitful and central research program measure.

6 Indeed, if µ0 is an ergodic invariant measure for the maps (maps of the circle or of the interval with a transformation T0, then the Ergodic Theorem ensures unique critical point) it is known that the above sce-

that for every µ-integrable function ϕ : M [ and nario holds true for Lebesgue almost every parame-

for µ-almost every point x in the manifold M the ter. It is known that there are systems admitting no

§ § §

1 n 1 j

¨ ¨£¨ time average ϕ˜ x lim n ϕ T x ex- physical measure, but the only known cases are not

n ∞ ∑ j 0 0 ists and equals the space average ϕdµ0. A phys- robust, i.e. there are systems arbitrarily close which ical measure µ is an invariant probability measure admit physical measures. for which it is required that time averages of every It is hoped that conclusions drawn from models ad- continuous function ϕ exist for a positive Lebesgue mitting physical measures to be effectively observ- measure (volume) subset of the space and be equal § able in the physical processes being modelled. In

to the space average µ ϕ ¨ . order to lend more weight to this expectation re- We note that if µ is a density, that is, is abso- searchers demand stability properties from such in- lutely continuous with respect to the volume mea- variant measures. sure, then the Ergodic Theorem ensures that µ is physical. However not every physical measure is absolutely continuous. To see why in a simple example Stochastic stability we just have to consider a singularity p of a vector field which is an attracting fixed point (a sink), then There are two main issues when we are given a mathematical model, both theoretical but with prac- the Dirac mass δp concentrated on p is a physical probability measure, since every orbit in the basin of tical consequences. The first one is to describe the

attraction of p will have asymptotic time averages for asymptotic behavior of most orbits, that is, to un-

§ §

derstand where do orbits go when time tends to in- ¨ any continuous observable ϕ given by ϕ p ¨ δp ϕ . finity. The second and equally important one is to Physical measures need not be unique or even ex- ascertain whether the asymptotic behavior is stable ist in general, but when they do exist it is desirable under small changes of the system, i.e. whether the that the set of points whose asymptotic time aver- limiting behavior is still essentially the same after ages are described by physical measures (such set small changes to the evolution law. In fact since is called the basin of the physical measures) be of models are always simplifications of the real system full Lebesgue measure — only an exceptional set of (we cannot ever take into account the whole state of points with zero volume would not have a well de- the universe in any model), the lack of stability con- fined asymptotic behavior. This is yet far from being siderably weakens the conclusions drawn from such proved for most dynamical systems, in spite of much models, because some properties might be specific to recent progress in this direction. it and not in any way resemblant of the real system. There are robust examples of systems admitting sev- Random dynamical systems come into play in this eral physical measures whose basins together are setting when we need to check whether a given of full Lebesgue measure, where robust means that model is stable under small random changes to the there are whole open sets of maps of a manifold in evolution law. the C2 topology exhibiting these features. For typical parameterized families of one-dimensional unimodal In more precise terms, we suppose that we are given a dynamical system (a transformation or a flow) ad-

7 mitting a physical measure µ0, and we take any measure on the circle, is T0-invariant and ergodic.

random dynamical system obtained from this one Then λ is physical. ]

through the introduction of small random perturba- ] 1 1

We consider the parameterized family Tt : §

tions on the dynamics, as in Examples 1- 4 or in ] 1

¨ 3 ¤

¤ t x x t and a family of probability mea-

§ § § §

Subsection 2.3, with the noise level ε 0 close to 1 ¢

N N

¨£¨ ¨£¨ ¤ sures θ λ ε ¤ ε λ ε ε given by the zero. ε normalized restriction of λ to the ε-neighborhood of ] 1 In this setting if, for any choice µε of invariant mea- 0, where we regard as the Lie group ^_G and use

sure for the random dynamical system for all ε # 0 additive notation for the group operation. Since λ is ] 1

small enough, the set of accumulation points of the Tt -invariant for every t , λ is also an invariant

§ ¨

family µε ε \ 0, when ε tends to 0 — also known as measure for the measure preserving random system § zero noise limits — is formed by physical measures ]

¨ ¤

or, more generally, by convex linear combinations of S : Ω λ θε ¤

§ § ]

physical measures, then the original unperturbed dy- 1

¨ ¨ for every ε 0, where Ω . Hence T ¤ λ is namical system is stochastically stable. 0 stochastically stable under additive noise perturba- This intuitively means that the asymptotic behavior tions. measured through time averages of continuous ob-

Concrete examples can be irrational rotations, §

servables for the random system is close to the be-

3 Rba

T0 x ¨ x α with α , or expanding maps of

§ ¥

havior of the unperturbed system.

Pc ¨ the circle, T0 x b x for some b ¤ n 2. Anal-

Recent progress in one-dimensional dynamics has ogous examples exist in higher dimensional tori.

¨ Y

shown that, for typical families f of maps of X t t 0 , 1 Example 7 (Stochastic stability depends on the type the circle or of the interval having a unique critical of noise). In spite of the straightforward way to ob-

point, a full Lebesgue measure subset T of the set of tain stochastic stability in Example 6, for e.g. an ex-

parameters is such that, for t T, the dynamics of ft ¢

panding circle map T0 x ¨ 2 x, we can choose a admits a unique stochastically stable (under additive continuous family of probability measures θε such noise type random perturbations) physical measure that the same map T0 is not stochastically stable. µt whose basin has full measure in the ambient space (either the circle or the interval). Therefore models It is well known that λ is the unique absolutely con- involving one-dimensional unimodal maps typically tinuous invariant measure for T0 and also the unique are stochastically stable. physical measure. Given ε # 0 small let us deﬁne transition probability measures as follows

In many settings (e.g. low dimensional dynamical

§ §

: N <

¨ ¨+3 systems) Lyapunov exponents can be given by time λ φ z ε ¤ φ z ε

ε ε

¢%

§ § § ¨

pε z ¤

: N <

¨ ¨+3 ¨

averages of continuous functions — for example the λ φε z ε ¤ φε z ε

0 0 § time average of log DT gives the biggest expo- §

0 ]

N N : <

¨6d R ¨ d ¤ where φε ε ¤ ε 0, φε 2ε 2ε T0 and

nent. In this case stochastic stability directly implies §

¨ ¤ stability of the Lyapunov exponents under small ran- over 2ε ¤ ε ε 2ε we deﬁne φε by interpolation

dom perturbations of the dynamics. in order that it be smooth.

N ¨ Example 6 (Stochastically stable examples). Let T0 : In this setting every random orbit starting at ε ¤ ε ] 1 ¦ be a map such that λ, the Lebesgue (length) never leaves this neighborhood in the future. More-

over it is easy to see that every random orbit eventu- entropy of µ with respect to ξ is deﬁned to be

N ¨

ally enters ε ¤ ε . Hence every invariant probabil-

§ § §

¨ ¨ ity measure µε for this Markov Chain model is sup- Hµ ξ ¨ ∑ µ R log µ R

R ξ

:gN < ported in ε ¤ ε . Thus letting ε 0 we see that the only zero-noise limit is δ the Dirac mass concen- 0 where we convention 0log 0 0. Given another ﬁ- trated at 0, which is not a physical measure for T0.

nite partition ζ we write ξ h ζ to indicate the partition This construction can be done in a random maps set- obtained through intersection of every element of ξ ting, but only in the C0 topology — it is not possible with every element of ζ, and analogously for any ﬁ- to realize this Markov Chain by random maps that nite number of partitions. If µ is also a stationary 1 are C close to T0 for ε near 0. measure for a random maps model (as in Subsec- tion 2.2), then for any ﬁnite measurable partition ξ of M,

n 1

§ § § 3.3 Characterization of measures satisfying §

i 1

( '

A lot of work has been put in recent years in extend- is ﬁnite and is called the entropy of the random dy-

ing important results from dynamical systems to the namical system with respect to ξ and to µ. §

random setting. Among many examples we mention

We define h sup h ξ ¨ as the metric entropy of the local conjugacy between the dynamics near a hy- µ ξ µ the random dynamical system, where the supremo perbolic fixed point and the action of the derivative is taken over all µ-measurable partitions. An impor- of the map on the tangent space, the stable/unstable tant point here is the following notion: setting A the manifold theorems for hyperbolic invariant sets and Borel σ-algebra of M, we say that a finite partition ξ the notions and properties of metric and topological of M is a random generating partition for A if entropy, dimensions and equilibrium states for po-

tentials on random (or fuzzy) sets. ∞

§ §

i 1 ¨ T ¨ ξ A The characterization of measures satisfying the En- ω

i 0

tropy Formula is one important result whose exten-

sion to the setting of iteration of independent and (except µ-null sets) for p -almost all ω Ω U . identically distributed random maps has recently had Then a classical result from Ergodic Theory ensures

interesting new consequences back into non-random that we can calculate the entropy using only a ran- § dynamical systems. dom generating partition ξ: we have hµ hµ ξ ¨ .

The Entropy Formula Metric entropy for random perturbations There exists a general relation ensuring that the en-

Given a probability measure µ and a partition ξ of tropy of a measure preserving differentiable transfor-

§ ¨ M, except perhaps for a subset of µ-null measure, the mation T0 ¤ µ on a compact Riemannian manifold is

9 bounded from above by the sum of the positive Lya- measures as zero-noise limits of random invariant

punov exponents of T0 measures in some settings, outlined in what follows,

§ §

§ obtaining in the process that the physical measures

;

¨ ¨ ¨

hµ T0 B ∑ λi x dµ x so constructed are also stochastically stable. Yj\ λi X x 0 The physical measures obtained in this manner ar- The equality (Entropy Formula) was first shown guably are natural measures for the system, since to hold for diffeomorphisms preserving a measure they are both stable under (certain types of) random equivalent to the Riemannian volume, and then the perturbations and describe the asymptotic behavior measures satisfying the Entropy Formula were char- of the system for a positive volume subset of ini- acterized: for C2 diffeomorphisms the equality holds tial conditions. This is a significant contribution to if, and only if, the disintegration of µ along the un- the state-of-the-art of present knowledge on Dynam- stable manifolds is formed by measures absolutely ics from the perspective of Random Dynamical Sys- continuous with respect to the Riemannian volume tems. restricted to those submanifolds. The unstable manifolds are the submanifolds of M everywhere tangent to the Lyapunov subspaces corresponding to all pos- Hyperbolic measures and the Entropy Formula itive Lyapunov exponents, the analogous to “inte- grating the distribution of Lyapunov subspaces cor- The main idea is that an ergodic invariant measure µ responding to positive exponents” — this particular for a diffeomorphism T0 which satisfies the Entropy point is a main subject of smooth ergodic theory for Formula and whose Lyapunov exponents are every- non-uniformly hyperbolic dynamics. where non-zero (known as hyperbolic measure) nec- Both the inequality and the characterization of sta- essarily is a physical measure for T0. This follows tionary measures satisfying the Entropy Formula from standard arguments of smooth non-uniformly were extended to random iterations of independent hyperbolic ergodic theory. and identically distributed C2 maps (non-injective Indeed µ satisfies the Entropy Formula if, and only if, and admitting critical points), and the inequality µ disintegrates into densities along the unstable sub- u §

reads manifolds of T0. The unstable manifolds W x ¨ are

§ §

§ tangent to the subspace corresponding to every pos-

;

BMB ¨ ¨ ¨ h λ x ¤ ω dµ x d p ω

µ ∑ i itive Lyapunov exponent at µ-almost every point x,

§ § §

Yj\ , λi X x ω 0

u u ¨ they are an invariant family, i.e. T0 W x ¨£¨ W x where the functions λi are the random variables pro- for µ-almost every x, and distances on them are uni- vided by the Random Multiplicative Ergodic Theo- 1 formly contracted under iteration by T0 . rem. If we know that the exponents along the complemen- tary directions are non-zero, then they must be nega- 3.4 Construction of physical measures as tive and smooth ergodic theory ensures that there ex- s §

zero-noise limits ist stable manifolds, which are submanifolds W x ¨ of M everywhere tangent to the subspace of negative

The characterization of measures which satisfy the Lyapunov exponents at µ-almost every point x, form

§ § §

s s ¨ Entropy Formula enables us to construct physical a T0-invariant family (T0 W x ¨£¨ W x , µ-almost

10 everywhere), and distances on them are uniformly Zero-noise limits satisfying the Entropy Formula contracted under iteration by T0. Using the extension of the characterization of mea- We still need to understand that time averages are sures satisfying the Entropy Formula for the random constant along both stable and unstable manifolds, maps setting, we can build random dynamical sys- and that the families of stable and unstable manifolds tems, which are small random perturbations of a map are absolutely continuous, in order to realize how an T , having invariant measures µ satisfying the En- hyperbolic measure is a physical measure. 0 ε

tropy Formula for all sufﬁciently small ε # 0. Indeed

s §

Given y W x ¨ the time averages of x and y co- it is enough to construct small random perturbations

incide for continuous observables simply because of T having absolutely continuous invariant proba-

§ § § 0

n n

¨ ¨£¨ 3 dist T0 x ¤ T0 y 0 when n ∞. For unsta- bility measures µε for all small enough ε 0. ble manifolds the same holds when considering time

1 In order to obtain such random dynamical systems

averages for T0 . Since forward and backward time we choose families of maps T : U M M and

averages are equal µ-almost everywhere, we see that § ¨ of probability measures θ \ as in Examples 3 the set of points having asymptotic time averages ε ε 0

and 4, where we assume that o U so that T0 be- §

given by µ has positive Lebesgue measure if the fol- §

¨ ¨

longs to the family. Letting Tx u T u ¤ x for all §

lowing set §

¨ ¨ u ¤ x U M, we then have that Tx θε is absolutely continuous. This means that sets of perturba-

tions of positive θε-measure send points of M onto

§ § §

lk s u positive volume subsets of M. This kind of perturba-

¨ ¨ B W y ¨ : y W x supp µ tion can be constructed for every continuous map of any manifold. In this setting we have that any invariant probabil- has positive volume in M, for some x whose time ity measure for the associated skew-product map

averages are well deﬁned.

¦ S : Ω M of the form θε µε is such that µε is Now, stable and unstable manifolds are transverse absolutely continuous with respect to volume on M. everywhere where they are deﬁned, but they are only Then the Entropy Formula holds

deﬁned µ-almost everywhere and depend measur-

§ § §

BmB ¨ ¨ ¨

ably on the base point, so we cannot use transver- hµε ∑ λi x ¤ ω dµε x dθε ω

Yj\ ,

sality arguments from differential topology, in spite λi X x ω 0

§ §

u § u

¨ ¨ of W x ¨ supp µ having positive volume in W x Having this and knowing the characterization of by the existence of a smooth disintegration of µ along measures satisfying the Entropy Formula, it is nat- the unstable manifolds. However it is known for ural to look for conditions under which we can guar- smooth (C2) transformations that the families of sta- antee that the above inequality extends to any zero- ble and unstable manifolds are absolutely continu-

noise limit µ of µ when ε 0. In that case µ sat- ous, meaning that projections along leaves preserve 0 ε 0 isﬁes the Entropy Formula for T . sets of zero volume. This is precisely what is needed 0 for measure-theoretic arguments to show that B has If in addition we are able to show that µ0 is a hyper- positive volume. bolic measure, then we obtain a physical measure for

11 T0 which is stochastically stable by construction. Products of random matrices Random maps These ideas can be carried out completely for hy- Random orbits perbolic diffeomorphisms, i.e. maps admitting an Random perturbations continuous invariant splitting of the tangent space Stochastic processes into two sub-bundles E n F defined everywhere with Stochastic differential equations bounded angles, whose Lyapunov exponents are Stochastic flows of diffeomorphisms negative along E and positive along F. Recently Stochastic stability maps satisfying weaker conditions where shown to admit stochastically stable physical measures following the same ideas. These ideas also have applications to the construc- Further Reading tion and stochastic stability of physical measure for strange attractors and for all mathematical models L. Arnold, (1998), Random dynamical systems. involving ordinary differential equations or iterations Springer-Verlag, Berlin. of maps. P. Billingsley, (1965), Ergodic theory and informa- tion. J. Wiley & Sons, New York. See also P. Billingsley, (1985), Probability and Measure. John Wiley and Sons, New York, 3rd edition. Equilibrium statistical mechanics J. Doob, (1953), Stochastic Processes. Wiley, New Dynamical systems York. Global analysis Non-equilibrium statistical mechanics A. Fathi, M. Herman, and J.-C. Yoccoz (1983), A Ordinary and partial differential equations proof of Pesin’s stable manifold theorem. In Geo- Stochastic methods metric dynamics (Rio de Janeiro 1981), volume 1007 Strange attractors of Lect. Notes in Math., pages 177–215. Springer Verlag, New York. Y. Kifer, (1986), Ergodic theory of random perturbations. Birkhauser¨ , Boston. Keywords Y. Kifer, (1988), Random perturbations of dynamical systems. Birkhauser¨ , Boston. Dynamical system H. Kunita, (1990), Stochastic flows and stochastic Flows differential equations, Cambridge University Press, Orbits Cambridge. Ordinary differential equations Markov chains F Ledrappier and L.-S. Young, (1998). Entropy for- Multiplicative Ergodic Theorem mula for random transformations. Probab. Theory Physical measures and Related Fields, 80(2): 217–240.

12 B. Øskendal, (1992), Stochastic Differential Equa- tions. Universitext. Springer-Verlag, Berlin, 3rd edition. P.-D. Liu and M. Qian, (1995) Smooth ergodic theory of random dynamical systems, volume 1606 of Lect. Notes in Math. Springer Verlag. P. Walters, (1982), An introduction to ergodic theory. Springer Verlag. C. Bonatti, L. D´ıaz, and M. Viana, (2004), Dynam- ics Beyond Hyperbolicity: A Global Geometric and Probabilistic perspective. Springer-Verlag, Berlin. M. Viana (2000). What’s new on Lorenz strange attractor. Mathematical Intelligencer, 22(3): 6–19.