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showed that a strongly causal spacetime which is

causally disconnected by a compact set contains a

nonspacelike geodesic line which intersects the

compact set. (Again, unless the spacetime is non-

spacelike geodesically complete, this line need not be

future or past complete.)

A pattern common to many results in global

Riemannian geometry especially since the 1950s is

the following: the existence of a complete Riemannian

metric on a smooth manifold which also satisfies a

global curvature inequality implies a topological or

geometric conclusion. A celebrated early example

from the 1950s and 1960s, obtained by separate

results of Rauch, Berger, and Klingenberg, is the

topological sphere theorem.

Topological Sphere Theorem Suppose (N, g

) is a

complete, simply connected Riemannian n-manifold

whose sectional curvatures satisfy 1=4 < K  1.

Then N is homeomorphic to S

By contrast, for spacetimes, the assumption of

geodesic completeness is generally unwarranted.

Here is an example of one of the celebrated

singularity theorems of general relativity, published

in 1970 as originally stated:

Hawking–Penrose Singularity Theorem No space-

time (M, g) of dimension n  3 can satisfy all of the

following three requirements together:

(i) (M, g) contains no closed timelike curves;

(ii) Every inextendible nonspacelike geodesic in

(M, g) contains a pair of conjugate points; and

(iii) There exists a future- or past-trapped set S in

(M, g).

This theorem may be reinterpreted more akin to

the Riemannian pattern above as follows: suppose

(M, g) is a chronological spacetime of dimensions

n  3 which satisfies the timelike convergence

condition (Ric(v, v)  0 for all timelike tangent

vectors) and the generic condition (every inextend-

ible nonspacelike geodesic contains a point which

has some appropriate nonzero sectional curvature).

If (M, g) contains a future- or past-trapped set, then

(M, g) is nonspacelike geodesically incomplete.

Hence, this result models the pattern: global

curvature inequalities (reflecting the physical

assumptions that gravity is assumed to be attractive

and every inextendible nonspacelike geodesic experi-

ences tidal acceleration) and a further physical or

geometric assumption (the first and third condit ions)

implies the existence of an incomplete timelike or

null geodesic.

An influential concept in global Riemannian

geometry formulated during the 1960s and 1970s

is that of curvature rigidity, which first became

widely known through the introduction to the text

Cheeger and Ebin (1975). The above statement of

the ‘‘sphere theorem’’ contains one hypothesis that

the sectional curvature is strictly greater than 1/4.

In curvature rigidity, the hypothesis of strict

inequality is relaxed to include the possibility of

equality as well, and then one tries to s how that

either the old conclusion is still valid, or if it fails, it

fails in an isometric (hence ‘‘rigid’’) manner. Thus

in the example of the sphere theorem, if the

sectional curvature is now allowed to satisfy 1=4 

K  1, then either the given Riemannian manifold

remains homeomorphic to the n-sphere, or if not, it

is isom etric to a Riemannian symmetric space of

rank 1.

Already in an article in 1970, Geroch had

expressed the opinion that most spacetimes should

be nonspacelike geodesically incomplete and also

that a spacetime should fail to be nons pacelike

geodesically incomplete only under special circum-

stances. Apparently by the early 1980s, S T Yau had

formulated the idea that timelike geodesic incom-

pleteness of spacetimes ought to display a curvature

rigidity. In the paragraph following the statement of

the Hawking–Penrose singularity theorem, there are

two curvature conditions mentioned – the timelike

convergence condition and the generic condition.

Now the timelike convergence condition already

allows for the case of equality (i.e., zero timelike

Ricci curvature) in its formulation; hence, curvature

rigidity here would imply dropping the generic

condition that each inextendible nonspacelike geo-

desic contains a point of nonzero sectional curva-

tures as a hypothesis. This notion seems first to have

been published by Yau’s Ph.D. student R Bartnik in

1988 as follows:

Conjecture Let (M, g) be a spacetime of dimension

3 which

(i) contains a compact Cauchy surface and

(ii) satisfies the timelike convergence condition

Ric(v, v)  0 for all timelike v.

Then either (M, g) is timelike geod esically incom-

plete, or (M, g) splits isometrically as a product

(jR  V, dt

þ h) where (H, h) is a compact

Riemannian manifold.

This conjecture has been proven in many cases

with the following proof scheme. From the physical

or geometric assumptions made, produce an

inextendible nonspacelike geodesic line. Further,

prove that the line happens to be timelike rather

than null. Then if the spacetime were timelike

geodesically complete, it would contain a complete

Lorentzian Geometry 347

timelike line. But then the desired splitting may be

obtained using the Lorentzian splitting theorem.

Lorentzian Splitting Theorem Let (M, g) be a

spacetime of dimension 3 which satisfies each of

the following conditions:

(i) (M, g) is either globally hyperbolic or timelike

geodesically complete;

(ii) (M, g) satisfies the timelike convergence condi-

tion; and

(iii) (M, g) contains a complete timelike line.

Then (M, g) splits isomet rically as a product (R  V,

dt

þ h) where (H, h) is a complete Riemannian

manifold.

This result, which corresponds to obtaining the

spacetime analog of a celebrated splitting theorem of

Cheeger and Gromoll for lines in complete Riemannian

manifolds of non-negative Ricci curvature, published

in 1971, was posed as a problem by S T Yau in a

problem list stemming from the conference

Special Year in Differential Geometry held at the

Institute for Advanced Study in Princeton during the

1979–80 academic year. Early progress was made

using maximal hypersurface methods by Gerhardt in

1983, Bartnik in 1984, and Galloway in 1984. Then

in 1985, Beem, Ehrlich, Markvorsen, and Galloway

introduced the methodology of employing the

Busemann function of the complete timelike line,

motivated by techniques from Riemannian geome-

try, and succeeded in obtaining a splitting under the

hypothesis of global hyperbolicity and everywhere

nonpositive timelike sectional curvatures. In separate

publications, Eschenb urg and Galloway extended

the result to the desired curvature hypothesis of

nonnegative timelike Ricci curvatures. Finally,

Newman in 1990 achieved the originally desired

goal of obtaining the splitting under the assumption

of timelike geodesic completeness, rather than global

hyperbolicity. This is a more delicate setting, since

timelike geodesic completeness does not imply

maximal nonspacelike geodesic connectability, a

fairly basic geometric tool in many standard

constructions. But the idea emerged with

Newman’s solution that the existence of a timelike

geodesic line or segment in a nonglobally hyper-

bolic spacetime implies an adequate level of control

in a tubular neighborhood of the given line to

enable the proof to work. Galloway and Horta in

1996 published a much simplified working out of

these concepts. A fuller exposition of these devel-

opments may be found in Beem et al. (1996,

chapter 14). In addition, in 2000, Galloway

published a version of the splitting theore m for a

null maximal geodesic line.

Two-Dimensional Spacetimes

Two-dimensional spacetimes, sometimes termed

Lorentz surfaces, are especially tractable because

given (M, g) with dim M = 2, then (M, g) is also a

spacetime. Hence, it may be shown that any

Lorentzian 2-manifold (M, g) homeomorphic to R

may be made geodesically complete (not just

nonspacelike geodesically complete) by a conformal

change of metric. Also, any simply connected two-

dimensional Lorentzian manifold is strongly causal.

In Weinstein (1996), an extensive study is made of

Lorentz surfaces generally and particularly, of a

conformal boundary for such surfaces first given by

Kulkarni in 1985.

One of the prettiest classical results linking the

geometry and topology of a Riemannian surface is

the Gauss–Bonnet theorem. Let (N, g

)bea

Riemannian manifold of dimension 2 and let P be a

polygonal subregion with piecewise smooth bound-

ing curves c

,1 i  k.LetK denote the Gauss

curvature of (N, g

) and  the geodesic curvature of

the smooth curves c

(which vanishes if c

happens to

be a geodesic). If 

denote the corresponding

interior angles between the successive boundary

curves c

and c

iþ1

, then the Gauss–Bonnet formula

over P is

K dA þ

 ds þ

ð  

Þ¼2 ½10

By considering a triangulation of N itself and

summing up the corresponding terms in [10],it

follows for a compact oriented Riemannian mani-

fold (N, g

) of dimension 2 that

K dA ¼ 2

ðNÞ

½11

where 

(N)

denotes the Euler characteristic. Also

lurking in the background here is a formula for

computing the angle between unit tangent vectors v,

w as

cos  ¼ g

ðv; wÞ½12

In the spacetime setting, different versions of a

Gauss–Bonnet formula for subregions of a two-

dimensional spacetime (M, g ) corresponding to [10]

have been given in 1974 by Helzer and in 1984 by

Birman and Nomizu. First, the angle computation is

a bit trickier for spacetimes than in the Riemannian

case; eqn [12] has to be replaced by techniques

which use the hyperbolic functions cosh u and

sinh u to define the angle u (sometimes called the

‘‘hyperbolic angle’’) between two unit vectors and

348 Lorentzian Geometry

then to allow for null vectors. Birman and Nomizu

obtained an analog of [10] assuming that the

boundary curves for P are successive smooth unit

timelike curves:

 ds 

K dA þ



¼ 0

Helzer in his formulation allows the different

boundary curves to be either unit timelike, unit

spacelike or null separately. Since the only compact,

orientable smooth surface which admits a spacetime

metric is the 2-torus, which has zero Euler char-

acteristic, the Riemannian formula [11] above

translates into the uniform constraint on the Gauss

curvature of the spacetime:

K dA ¼ 0

See also: General Relativity: Overview; Geometric

Analysis and General Relativity; Pseudo-Riemannian

Nilpotent Lie Groups; Spacetime Topology, Causal

Structure and Singularities.

Further Reading

Beem J, Ehrlich P, and Easley K (1996) Global Lorentzian

Geometry. In: Marcel Dekker Pure and Applied Mathematics,

vol. 202, 2nd edn. New York: Dekker.

Cheeger J and Ebin D (1975) Comparison Theorems in

Riemannian Geometry, North Holland Mathematical Library.

Amsterdam: North-Holland.

Einstein A (1916) Die Grundlage der allgemeinen Relativita¨ tsthe-

orie. Annalen der Physik 49: 769–822.

Gromov M (1999) Metric Structures for Riemannian and Non-

Riemannian Spaces, Birkhauser Progress in Mathematics,

vol. 152. Boston: Birkhauser.

Hawking S and Ellis G (1973) The Large Scale Structure of

Spacetime. Cambridge: Cambridge University Press.

Karcher H (1989) Riemannian comparison constructions. In:

Chern SS (ed.) Global Differential Geometry,Mathematical

Association of America Studies in Mathematics, vol. 27,

pp. 170–222. Washington, DC: MAA.

Kriele M (1999) Spacetime: Foundations of General Relativity

and Differential Geometry, Springer Lecture Notes in Physics

(Monographs), vol. 59. Heidelberg: Springer.

Noldus J (2004) The limit space of a Cauchy sequence of globally

hyperbolic spacetimes. Classical Quantum Gravity 21:

851–874.

O’Neill B (1983) Semi-Riemannian Geometry with Applications

to Relativity, Academic Press Pure and Applied Mathematics.

New York: Academic Press.

Penrose R (1972) Techniques of Differential Topology in

Relativity. Society for Industrial and Applied Mathematics,

Regional Conference Series in Applied Mathematics, vol. 7.

Philadelphia: SIAM.

Petersen P (1998) Riemannian Geometry, Springer Verlag Gradu-

ate Texts in Mathematics, vol. 171. New York: Springer.

Sachs R and Wu H (1977) General Relativity for Mathematicians,

Springer Verlag Graduate Texts in Mathematics, vol. 48. New

York: Springer.

Weinstein T (1996) An Introduction to Lorentz Surfaces,de

Gruyter Expositions in Mathematics, vol. 22. Berlin: de

Gruyter.

Lyapunov Exponents and Strange Attractors

M Viana, IMPA, Rio de Janeiro, Brazil

Lyapunov Exponents

The Lyapunov exponents of a sequence {A

, n 1}

of square matrices of dimension d 1 are the values

ðvÞ¼lim sup

n!1

log kA

 vk½1

over all nonzero vectors v 2 R

. For completeness,

set (0) = 1. It is easy to see that (cv) = (v) and

(v þ v

)  max{(v), (v

)} for any nonzero scalar c

and any vectors v, v

. It follows that, given any

constant a, the set of vectors satisfying (v) a is a

vector subspace. Consequently, there are at most d

Lyapunov exponents, henceforth denoted by



<  <

k1

<

, and there exists a filtration

<  < F

k1

< F

= R

into vector subspaces,

such that

ðvÞ¼

for all v 2 F

i1

and every i = 1, ..., k (write F

= {0}). In particular,

the largest exponent is



¼ lim sup

n!1

log kA

k½2

One calls dim F

 dim F

i1

the multiplicity of each

Lyapunov exponent 

There are corresponding notions for continuous

families of matrices A

, t 2 (0, 1), taking the limit as

t goes to 1 in the relations [1] and [2]. The theories

for the two types of families, discrete and contin-

uous, are analogous and so at each point in what

follows we refer to either one or the other.

Lyapunov Exponents and Strange Attractors 349

Lyapunov Stability

Consider the linear differential equation

vðtÞ¼BðtÞvðtÞ½3

where B(t) is a bounded function with values in the

space of d  d matrices, defined for all t 2 R. The

theory of differential equations ensures that there

exists a fundamental matrix A

, t 2 R, such that

vðtÞ¼A

 v

is the unique solution of [3] with initial condition

v(0) = v

If the Lyapunov exponents of the family A

, t > 0,

are all negative then the trivial solution v(t )  0is

asymptotically stable, and even exponentially stable.

The stability theorem of Lyapunov asserts that,

under an additional regularity condition, stability is

still valid for nonlinear perturbations

wðtÞ¼BðtÞw þ Fðt; wÞ½4

with kF(t, w)kconst.kwk

1þc

, c > 0. That is, the

trivial solution w(t)  0 is still exponentially asymp-

totically stable.

The regularity condition means, essentially, that

the limit in [1] does exist, even if one replaces

vectors v by elements v

^^v

of any lth exterior

power of R

,1l d. By definition, the norm of an

l-vector v

^^v

is the volume of the parallele-

piped determined by the vectors v

, ..., v

. This

condition is usually tricky to check in specific

situations. However, the multiplicative ergodic

theorem of V I Oseledets asserts that, for very

general matrix-valued stationary random processes,

regularity is an almost sure property. This result sets

the foundation for the modern theory of Lyapunov

exponents. We are going to discuss the precise

statement of the theorem in the slightly broader setting

of linear cocycles, or vector bundle morphisms.

Linear Cocycles

Let  be a probability measure on some space M and

f : M !M be a measurable transformation that

preserves . Let  : E!M be a finite-dimensional

vector bundle, endowed with a Riemannian metric

kk

on each fiber E

= 

1

(x). Let A: E!E be a

linear cocycle over f. What we mean by this is that

 A¼f  

and the action A(x):E

f (x)

of A on each fiber is

a linear isomorphism. Notice that the action of the

nth iterate A

is given by

ðxÞ¼Aðf

n1

ðxÞÞAðf ðxÞÞ  AðxÞ

for every n 1.

Assume the function log

kA(x)k

is -integrable:

log

kAðxÞk

2 L

ðÞ½5

(we write log

 = log max {, 1}, for any >0).

It is clear that the sequence of functions

(x) = log kA

(x)k

satisfies

mþn

ðxÞa

ðxÞþa

ðf

ðxÞÞ

for every m, n, and x. It follows from J Kingman’s

subadditive ergodic theorem that the limit

lim

n!1

ðxÞ

exists for -almost all x. In view of [2], this means

that the largest Lyapunov exponent 

(x) of the

sequence A

(x), n 1 is a limit, and not just a lim

sup, at almost every point.

Multiplicative Ergodic Theorem

The Oseledets theorem states that the same holds

for all Lyapunov exponents. Namely, for -almost

every x 2 M there exists k = k(x) 2 {1, ..., d}, a

filtration

<  < F

k1

< F

¼E

and numbers 

(x) <  <

(x) such that

lim

n!1

log kA

ðxÞk

¼ 

ðxÞ½6

for all v 2 F

i1

and i 2 {1, ..., k}.

The Lyapunov exponents 

(x), and their number

k(x), are measurable functions of x and they are

constant on orbits of the transformation f.In

particular, if the measure  is ergodic then k and

the 

are constant on a full -measure set of

points. The subspaces F

also depend measurably

on the point x and are invariant under the linear

cocycle:

AðxÞF

¼ F

f ðxÞ

It is in the nature of things that, usually, these

objects are not defined everywhere and they depend

discontinuously on the base point x.

When the transformation f is invertible, one

obtains a stronger conclusion, by applying the

previous kind of result also to the inverse of the

cocycle. Namely, assuming that log

1

k is also

in L

(), one gets that there exists a

decomposition

¼ E

E

350 Lyapunov Exponents and Strange Attractors

defined at almost every point and such that

A(x)  E

= E

f (x)

and

lim

n!1

log kA

ðxÞk

¼ 

ðxÞ½7

for all v 2 E

different from zero and all i 2

{1, ..., k}. These Oseledets subspaces E

are related

to the subspaces F

through

i¼1

Hence, dim E

= dim F

 dim F

i1

is the multipli-

city of the Lyapunov exponent 

(x).

The angles between any two Oseledets subspaces

decay subexponentially along orbits of f:

lim

n!1

log angle E

ðxÞ

; E

ðxÞ



¼0

for every i 6¼ j and almost every point. These facts

imply the regularity condition mentioned previously

and, in particular,

lim

n!1

log jdet A

ðxÞj ¼

i¼1



ðxÞdim E

½8

Consequently, for cocycles with values in SL(d, R),

the sum of all Lyapunov exponents, counted with

multiplicity, is identically zero.

As we are dealing with almost certain properties,

we may generally restrict the vector bundle to some

full measure subset over which it is trivial. Then

each fiber E

is identified with the space R

, and we

may think of A(x)asad  d matrix. Then

(x) = A(f

(x)) is a stationary random process

relative to (f, ). Thus, in this context it is no

serious restriction to view a linear cocycle as a

stationary random process with values in the linear

group GL(d, R) of invertible d  d matrices.

Furthermore, given any such random process

, n 0, one may consider its normalization

= A

=jdetA

j. The Lyapunov exponents of the

two random processes A

, n 0, and B

, n 0, differ

by the time average

lim

n!1

n1

j¼0

log jdetA

ðxÞj

of the determinant. The Birkhoff ergodic theorem

ensures that the time average is well defined almost

everywhere, as long as the function log jdet Aj is in

(); this is the case, for instance, if both

log

1

k are integrable. This relates the general

case to random processes with values in the special

linear group SL(d, R)ofd  d matrices with

determinant 1.

The Oseledets theorem was extended by D Ruelle

to certain linear cocycles in infinite dimensions. He

assumes that the A(x) are compact operators on a

Hilbert space H and log

kAk is in L

(). The

conclusion is the same as in finite dimensions,

except that the filtration

< F

< < F

¼ H

may involve infinitely many subspaces, and the

Lyapunov exponents may be 1. There is also a

version for cocycles over invertible transforma-

tions, where one assumes each A(x) to be invertible

and the sum of a unitary operator with a compact

operator, such that both log kA



k are integrable.

The conclusion is that there exists an Oseledets

decomposition H = E

E

 at almost

every point, with finitely or countably many

factors.

Random Matrices

Relation [8] implies that, for SL(d, R) cocycles, if

there is only one Lyapunov exponent (with full

multiplicity) then it must be zero. When this

happens, the theory contains no information on the

behavior of the iterates A

(x)  v, apart from the fact

that there is no exponential growth nor decay of

their norms. Thus, the question naturally arises

under which conditions is there more than one

Lyapunov exponent or, equivalently, under which

conditions is the largest Lyapunov exponent strictly

positive.

This problem was first addressed by H Furstenberg

for products of independent random variables,

corresponding to the following class of linear

cocycles. Let  be a probability measure on the

group G = GL(d, R). Consider M = G

and  = 

(or M = G

and  = 

), and let f : M !M be the

shift map



ð



¼ð

jþ1

It is clear that  is invariant and also ergodic for the

transformation f. Consider the cocycle A: E!E

defined by E = M  R

and



ð



 v ¼ 

 v

Clearly,



ð



 v ¼ 

n1





 v

Corresponding to the hypothesis of the multiplicative

ergodic theorem, assume that log

kk (and

log

k

1

k)are-integrable functions of the matrix .

Furstenberg’s theorem states that if the closed

group G () generated by the support of  is

Lyapunov Exponents and Strange Attractors 351

noncompact and strongly irreducible in R

then

the largest Lyapunov exponent of the cocycle A

is strictly positive. Strong irreducibility means

that there exists no finite union of subspaces of

that is invariant under all elements of the

group. Improvements, extensions, and alternative

proofs have been obtained by several authors since

then.

Especially, Y Guivarc’h and A Raugi provided

conditions under which there are exactly d distinct

Lyapunov exponents or, in other words, the

multiplicity of every Lyapunov exponent is equal

to 1. A matrix semigroup has the contraction

property if there exists a sequence of elements h

and a probability measure on the projective space

of R

that gives zero weight to any projective

subspace, such that the images (h

)



m of m under

the h

converge to a Dirac mass in the projective

space. They proved that if the closed semigroup

H() generated by the support of the probability 

is strongly irreducible and has the contraction

property then the largest Lyapunov exponent has

multiplicity 1. Applying this to the exterior

powers of the cocycle, one obtains sufficient

conditions for simplicity of the other Lyapunov

exponents as well.

This statement has been improved by I Ya

Gol’dsheid and G A Margulis, who formulated the

hypotheses in terms of the algebraic closure

G()of

the semigroup H(). They assumed that

G() has the

contraction property and the connected component

of the identity inside

G() is irreducible in R

meaning that its elements do not have any common

invariant subspace. Then the largest Lyapunov

exponent is simple.

Schro¨ dinger Cocycles

The one-dimensional discrete Schro¨ dinger equation

is the second-order difference equation

ðu

nþ1

þ u

n1

ÞþV

¼ Eu

½9

derived from the stationary Schro¨ dinger equation in

dimension 1 by space discretization. Here the energy

E is a constant and V

= V(f

()), where the

potential V() is a bounded scalar function and

f : M !M is a transformation preserving some

probability measure  on M. In what follows, we

take  to be ergodic. Equation [9] may be rewritten

as a first-order relation,

nþ1



 E 1



Hence, it may also be interpreted as a linear cocycle

A over f, where the vector bundle is E = M  R

and

AðÞ¼

VðÞE 1



½10

takes values in SL(R, 2). By ergodicity, the Lyapu-

nov exponents are essentially independent of the

base point . Let (E) denote the largest exponent:

by the relation [8], the other one is (E).

The Lyapunov exponent (E) is related to the

spectral theory of the linear operators L



ðL



uÞ

¼ðu

nþ1

þ u

n1

ÞþV

on the space ‘

(Z) of complex square-integrable

sequences u

, n 2 Z. These are bounded Hermitian

operators and so the spectra are compact subsets of R.

Using the assumption that  is ergodic, one can prove

that the spectrum spec(L



)isconstantalmostevery-

where. If the transformation f is minimal, the spectrum

is even independent of the point .Moreover,forall

energies,

ðEÞconst: distðE; specðL



ÞÞ

In particular, (E) is always positive on the comple-

ment of the spectrum.

A fundamental problem (Anderson localization) is

to decide when the spectrum is pure-point. This is

reasonably well understood for a few classes of base

dynamics only, for example, the very chaotic systems

such as Bernoulli and Markov processes (random

potentials) or uniformly hyperbolic maps and flows,

or the irrational rotations on the d-dimensional torus

(quasiperiodic potentials). In the latter case, the

results are more complete when there is only one

frequency (d = 1). It was shown by K Ishii and by L

Pastur that if (E) is positive for almost all values of

E in some Borel set then the absolutely continuous

part of the spectrum is essentially disjoint from that

set. The converse is also true (due to S Kotani). Thus,

checking that (E) is positive is an important step

towards proving localization.

A very general criterion for positivity of the

Lyapunov exponent was obtained by Kotani. Namely,

he proved that if the potential is not deterministic then

(E) is positive for almost all E. In particular, for

nondeterministic potentials the absolutely continuous

spectrum is empty, almost surely. In simple terms, the

hypothesis means that from the values of the potential

for negative n one cannot determine the values for

positive n. More formally, one calls the potential

deterministic if every V

, n 0 is almost everywhere a

measurable function of {V

: n 0}. For instance,

quasiperiodic potentials are deterministic, whereas

Bernoulli potentials are not.

352 Lyapunov Exponents and Strange Attractors

Subharmonicity Method

Let D

be the set of complex vectors (z

, ..., x

) 2 C

such that jz

j1forallj and let T

be the subset

defined by jz

j= 1forallj.Letf : T

and

A : T

!SL(d , R) be continuous maps that admit

holomorphic extensions to the interior of D

with

f (0) = 0. Assume that f preserves the natural (Haar)

measure  on T

.Let

ðA;Þ¼

ðzÞd

where (z) denotes the largest Lyapunov exponent

for the cocycle defined by A over f. It also follows

from the subadditive ergodic theorem that

ðA;Þ¼lim

log kA

ðzÞkd

M Herman observed that, since the function

log kA

(z)k is plurisubharmonic on D

, one may

use the maximum principle to conclude that

log kA

ðzÞkd 

log kA

ð0Þk

Then, taking the limit when n !1one obtains that

ðA;ÞðAÞ½11

where  (A) denotes the spectral radius of the matrix

A(0). Starting from this observation, he developed a

very effective method for bounding Lyapunov

exponents from below, that received several applica-

tions and extensions, in particular, to the theory of

Schro¨ dinger cocycles with quasiperiodic potentials.

The best-known application is the following bound

for integrated Lyapunov exponents of two-dimen-

sional cocycles. Let f : M !M be a continuous

transformation on a compact metric space, preserving

some probability measure ,andA : M ! SL(2, R)be

a continuous map. For each fixed ,letAR



be the

cocycle obtained by multiplying A(x), at every point

x, by the rotation of angle . Herman proved that

2

ðAR



;Þd 

NðxÞd

(A Avila and J Bochi later showed that the equality

holds) where

NðxÞ¼log

kAðxÞk þ kAðxÞ

1

Apart from the exceptional case when A acts by

rotation at every point in the support of , the right-

hand side of the inequality is positive, and so the

Lyapunov exponent of the cocycle AR



is positive

for many values of .

Nonuniform Hyperbolicity

The prototypical example of a linear cocycle is the

derivative of a smooth transformation on a mani-

fold. More precisely, let M be a finite-dimensional

manifold and f : M !M be a diffeomorphism, that

is, a bijective smooth map whose derivative Df (x)

depends continuously on x and is an isomorphism at

every point. Let E = TM be the tangent bundle to the

manifold and A= Df be the derivative. If M is

compact or, more generally, if the norms of both Df

and its inverse are bounded, then the hypothesis in

Oseledets theorem is automatically satisfied for any

f-invariant probability . Lyapunov exponents yield

deep geometric information on the dynamics of the

diffeomorphism, especially when they do not vanish.

For most results that we mention in the sequel, one

needs the derivative Df to be Ho¨ lder continuous:

kDf ðxÞDf ðyÞk  const: dðx; yÞ

Let E

be the sum of the Oseledets subspaces

corresponding to negative Lyapunov exponents.

Pesin’s stable manifold theorem states that there

exists a family of embedded disks W

loc

(x) tangent to

at almost every point and such that the orbit of

every y 2 W

loc

(x) is exponentially asymptotic to the

orbit of x. This lamination {W

(x)} is invariant, in

the sense that

f ðW

ðxÞÞ  W

ðf ðxÞÞ

and has an ‘‘absolute continuity’’ property. There

are analogous results for the sum E

of the Oseledets

subspaces corresponding to positive Lyapunov

exponents.

The entropy of a partition P of M is defined by



ðf ; PÞ ¼ lim

n!1



ðP

where P

is the partition into sets of the form

P = P

\ f

1

) \\f

n

) with P

2Pand



ðP

Þ¼

P2P

ðPÞlog ðPÞ

The Kolmogorov–Sinai entropy h



(f ) of the system

is the supremum of h



(f , P) over all partitions P

with finite entropy. The Ruelle–Margulis inequality

says that h



(f ) is bounded above by the average sum

of the positive Lyapunov exponents. A major result

of the theory, Pesin’s entropy formula, asserts that if

the invariant measure  is smooth (e.g., a volume

element) then the two invariants coincide:



ðf Þ¼

j¼1



d

Lyapunov Exponents and Strange Attractors 353

A complete characterization of the invariant mea-

sures for which the entropy formula is true was

given by F Ledrappier and L S Young.

The invariant measure  is called hyperbolic if all

Lyapunov exponents are nonzero at almost every

point. Hyperbolic measures are exact dimensional:

the pointwise dimension

dðxÞ¼lim

r!0

log ðB

ðxÞÞ

log r

exists at almost every point, where B

(x) is the

neighborhood of radius r around x. This fact was

proved by L Barreira, Ya Pesin, and J Schmeling. Note

that it means that the measure (B

(x)) of neighbor-

hoods scales as r

d(x)

when the radius r is small.

Another remarkable feature of hyperbolic mea-

sures, proved by A Katok, is that periodic motions

are dense in their supports. More than that,

assuming the measure is nonatomic, there exist

Smale horseshoes H

with topological entropy

arbitrarily close to the entropy h



(f ) of the system.

In this context, the topological entropy h(f , H

) may

be defined as the exponential rate of growth,

lim

k!1

log #fx 2 H

: f

ðxÞ¼xg

of the number of periodic points on H

Generic Systems

Given any area-preserving diffeomorphism on any

surface M, one may find another whose first

derivative is arbitrarily close to the initial one and

which has Lyapunov exponents identically zero at

almost every point, or else is globally uniformly

hyperbolic (Anosov). This surprising fact was

discovered by R Man˜e´, and a complete proof was

given by J Bochi. Uniform hyperbolicity means that

the tangent bundle admits a Df-invariant splitting

TM ¼ E

 E

such that the line bundle E

is uniformly contracted

and E

is uniformly expanded by the derivative. It is

well known that Anosov diffeomorphisms can only

occur if the surface is the torus T

In fact, the theorem of Man˜e´–Bochi is stronger:

for a residual subset (a countable intersection of

open dense sets) of all once-differentiable area-

preserving diffeomorphisms on any surface, either

the Lyapunov exponents vanish almost everywhere

or the diffeomorphism is Anosov. This shows that

zero Lyapunov exponents are actually quite com-

mon for surface diffeomorphisms that are only once-

differentiable. Moreover, this theorem has been

extended to diffeomorphisms on manifolds with

arbitrary dimension, in a suitable formulation, by

J Bochi and M Viana.

However, this phenomenon should be specific to

systems with low differentiability. Indeed, already

for Ho¨ lder-continuous linear cocycles over chaotic

transformations it is known that vanishing Lyapu-

nov exponents can only occur with infinite codimen-

sion. That is, unless the cocycle satisfies an infinite

number of independent constraints, there exists

some positive exponent. By ‘‘chaotic’’ we mean

here that the invariant probability  of the base

transformation is assumed to be hyperbolic and to

have local product structure: it is locally equivalent

to a product of two measures, respectively, along

stable and unstable sets.

Under additional assumptions, one can even prove

that all Lyapunov exponents have multiplicity 1

outside an infinite-codimension subset. This follows

from extensions of the Guivarc’h–Raugi criterion for

certain linear cocycles over chaotic transformations,

obtained by A Avila, C Bonatti, and M Viana.

Strange Attractors

This expression was coined by D Ruelle and

F Takens in their celebrated study on the nature of

fluid turbulence. E Hopf and also L D Landau and

E M Lifshitz had suggested that turbulent motion

arises from the existence in the phase space of

invariant tori carrying quasiperiodic flows with

large number of frequencies. Ruelle and Takens

observed that dissipative systems such as viscous

fluids do not generally have such quasiperiodic tori,

and concluded that turbulence must be credited to a

different mechanism: the presence of some ‘‘strange’’

attractor.

While they did not propose a precise definition,

two main features were mentioned:

1. Complex geometry: a strange attractor is not

reduced to an equilibrium point or a periodic

solution of the system and, generally, should

have a fractal structure.

2. Chaotic dynamics: solutions accumulating on the

attractor should be sensitive to their initial states.

As more examples were found, it became appar-

ent that the above two features do not always come

together. This led to two types of definitions in the

literature, depending on whether one emphasizes the

geometry or the dynamics. We adopt the second

point of view, and propose to define the strange

attractor as one carrying an invariant ergodic

physical measure which has some positive Lyapunov

exponent. The notion of physical measure will be

354 Lyapunov Exponents and Strange Attractors

defined near the end. The condition on the Lyapu-

nov exponent ensures that the dynamics near the

attractor is (exponentially) sensitive to the initial

states.

Lorenz-Like Attractors

The uniformly hyperbolic attractors introduced by

S Smale provided an interesting class of examples of

strange attractors, both chaotic and fractal. Perhaps

more striking, given that they originated from a

concrete problem in fluid dynamics, were the

strange attractors introduced by E N Lorenz. The

Lorenz system of differential equations,

x ¼x þ  y;¼ 10

y ¼ rx  y  xz; r ¼ 28

z ¼ xy  bz; b ¼ 8=3

½12

was derived from Lord Rayleigh’s model for

thermal convection, by Fourier expansion of the

stream function and temperature, and truncation of

all but three modes. Lorenz observed that its

solutions depend sensitively on their initial states.

Consequently, predictions based on the numerical

integration of the equations may turn out to be

very inaccurate, given that the initial data obtained

from experimental measurements are never com-

pletely precise. This remarkable observation

brought the issue of predictability in deterministic

systems to a whole new light and motivated intense

investigation of this and many other chaotic

systems.

The dynamical behavior of the eqns [12] was first

interpreted through certain geometric models where

the presence of strange attractors, both chaotic and

fractal, could be proved rigorously. It was much

harder to prove that the original eqns [12] them-

selves have such an attractor. This was achieved just

a few years ago, by W Tucker, by means of a

computer-assisted rigorous argument. At about the

same time, a mathematical theory of Lorenz-like

attractors in three-dimensional space was developed

by C Morales, M J Pacifico, and E Pujals. In

particular, this theory shows that uniformly hyper-

bolic attractors and Lorenz-like attractors are the

only ones which are robust under all small mod-

ifications of the vector field.

He´ non-Like Attractors

Starting from the work of Lorenz, many models of

strange attractors have been found and described to

some extent, often related to concrete problems.

From a mathematical point of view, it is usually

hard to give even a rough description of the

dynamics in the chaotic regime. However, this was

especially successful for the family of strange

attractors introduced by M He´non. He considered

a very simple nonlinear system, particularly suited

for numerical experimentation: the transformation

f ðx; yÞ¼ð1  ax

þ by; xÞ½13

where a and b are constant parameters. In a

breakthrough, M Benedicks and L Carleson were

able to prove that, for a set of parameter values with

positive probability, this transformation has some

nonhyperbolic attractor such that the orbits accu-

mulating on it are sensitive to the starting point. The

system [13] is also a model for many other

situations, including the phenomenon of creation of

homoclinic motions as parameters unfold, and the

conclusions of Benedicks and Carleson have been

extended to such situations, starting from the work

of L Mora and M Viana.

Moreover, a detailed theory of He´non-like attrac-

tors has been developed by M Benedicks, M Viana,

D Wang, L S Young, and other authors. It follows

from this theory that these attractors carry an

invariant ergodic probability measure  which

describes the statistical behavior of almost all

trajectories f

(x), j  1, that accumulate the

attractor:

lim

n!1

j¼1

’ðf

ðxÞÞ ¼

’ d

for any continuous function ’. This property

implies that, despite the fact that it is supported

on a zero-volume set, the measure  is, in some

sense, physically observable. For this reason, one

calls it a physical measure. In other words, time

averages along typical orbits in the domain of

attraction coincide with the space averages deter-

mined by the probability . Another property with

physical relevance is that  is the zero-noise limit of

the stationary measures associated to the Markov

chains obtained by adding random noise to f. One

says that the system (f , ) is stochastically stable.

See also: Chaos and Attractors; Dissipative Dynamical

Systems of Infinite Dimension; Ergodic Theory; Fractal

Dimensions in Dynamics; Generic Properties of

Dynamical Systems; Gravitational N-Body Problem

(Classical); Homoclinic Phenomena; Hyperbolic

Dynamical Systems; Lagrangian Dispersion (Passive

Scalar); Nonequilibrium Statistical Mechanics: Interaction

between Theory and Numerical Simulations; Random

Dynamical Systems; Synchronization of Chaos.

Lyapunov Exponents and Strange Attractors 355