Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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degenerate. The aim, if possible, is to find an explicit

gauge function so that the corresponding generalized

Minkowski contents or Hausdorff measure of A be

nondegenerate.

Methods of Fractal Analysis in Dynamics

Thermodynamic Formalism

Thermodynamic formalism has been developed by

Sinai (1972), Ruelle (1973), and Bowen (1975),

using methods of statistical mechanics in order to

study dynamics and to find dimensions of various

fractal sets. We first describe a ‘‘dictionary’’ for

explicit geometric constructions of Cantor-like sets.

Let X

be the set of all sequences i = (i

, i

, ...)of

elements i

from a given set of p symbols, say

{1, 2, ..., p}. We endow X

with the metric d(i, j):=

k

 j

j and introduce the one-sided shift

operator (or left shift)  : X

defined by

((i))

= i

nþ1

, that is, (i

, i

, ...) = (i

, i

, ...).

A set Q ˝ X

is called the symbolic dynamics if it is

compact and -invariant, that is, (Q) ˝ Q. Hence,

(Q, ) is a symbolic dynamical system. Denote

i[n]:= (i

, ..., i

). Given a continuous function

’ : Q !R, let us define the topological pressure of

’ with respect to  by

Pð’Þ:¼ lim

n!1

log

fi½n:i 2Qg

Eði½nÞ

Eði½nÞ:¼ exp sup

fj 2Q: j½n¼i½ng

n1

k¼0

’ð

ðjÞÞ

The topological entropy of  jQ is defined by

h(jQ):= P(0), that is,

hðjQÞ¼ lim

n!1

log #fi½n: i 2Qg

where # denotes the cardinal number of a set. The

above function ’

n1

k = 0

’  

has the property

’

nþm

= ’

þ ’

 

, and therefore we speak about

additive thermodynamic formalism. Topological pres-

sure was introduced by D Ruelle (1973) and extended

by P Walters (1976). Bowen’s equation (1979) has a

very important role in the computation of the

Hausdorff dimension of various sets. For the unknown

s 2R, and with a suitably chosen function ’,this

equation reads

Pðs’Þ¼0

Geometric Constructions

A geometric construction (Q, )inR

indexed by

symbolic dynamics Q is a family  of compact sets



i[n]

R

, i 2Q, n 2N, such that diam

i[n]

!0as

n !1, 

i[nþ1]

˝ 

i[n]

, 

i[n]

= int

i[n]

for every i 2Q

and all n, and int

i[n]

\ int

j[n]

= ; whenever

i[n] 6¼ j[n] (Moran’s open set condition). This family

induces the Cantor-like set

F :¼

n¼1

[

i 2Q



i½n

(see Figure 2). The mapping h : Q !F defined by

h(i):= \

n = 1



i[n]

is called the coding map of F.The

above geometric construction includes well-known

iterated function systems of similarities as a special

case. If 

, ..., 

are given numbers in (0, 1), and 

i[n]

are balls of radii r

i[n]

:= 

...

,thens := dim

F is

the unique solution of Bowen’s equation P(s’) = 0,

where ’ is defined by ’(i):= log 

(Ya Pesin and

H Weiss, 1996). In this case Bowen’s equation is

equivalent to Moran’s equation (1946),

k¼1



¼1

This result has been generalized by L Barreira (1996)

using the Carathe´odory–Pesin construction (1988).

Let us illustrate Barreira’s theory of nonadditive

thermodynamic formalism with a special case.

Assume that (Q, ) is a geometric construction for

which the sets 

i[n]

are balls, and let there exist >0

such that r

i[nþ1]

   r

i[n]

and r

i[nþm]

 r

i[n]



(i)[m]

for

all i 2Q, n, m 2N. Then dim

F = dim

F = s, where

s is the unique real number such that

lim

n!1

log

fi½n:i 2Qg

i½n

¼0 ½1

This is a special case of Barreira’s extension of

Bowen’s equation to nonadditive thermodynamic

formalism. Moran’s equation can be deduced from

[1] by defining r

i[n]

:= 

...

, where i = (i

, i

, ...),

and 

, ..., 

2 (0, 1) are given numbers. Pesin and

Weiss (1996) showed that Moran’s open set condi-

tion can be weakened so that partial intersections of

interiors of pairs of basic sets in the family  are

allowed. Thermodynamic formalism has been used

to study the Hausdorff dimension of Julia sets

Figure 2 Cantor-like set.

396 Fractal Dimensions in Dynamics

(Ruelle, 1982), horseshoes (H McCluskey and

A Manning, 1983), etc.

An important example of symbolic dynamics is the

topological Markov chain X

generated by a p  p

matrix A with entries a

2{0, 1}:

:¼fi ¼ði

; i

; ...Þ2X

: a

kþ1

¼1 for all k 2Ng

It is a compact, -invariant subset of X

.Themap

 jX

is called the subshift of finite type (Bowen,

1975). A construction of Cantor-like set F using

dynamics Q = X

is called a simple geometric con-

struction, while a geometric construction is said to be a

Markov geometric construction if Q = X

.IfF is

obtained by a Markov geometric construction such

that all 

i[n]

are balls of radii r

i[n]

:= 

...

, where



2(0, 1), i

2{1, ..., p}, then dim

F = dim

F = s,

where s is the unique solution of equation

(AM

) = 1. Here M

:= diag(

, ..., 

)and

(AM

) is the spectral radius of the matrix AM

.This

and more general results have been obtained by Pesin

and Weiss (1996).

Any Cantor-like set F obtained via iterated

function system of similarities satisfying Moran’s

open set condition is Hausdorff nondegenerate

(Moran, 1946). If F is of nonlattice type, that is,

the set { log 

, ..., log 

} is not contained in r  Z

for any r > 0, then F is Minkowski measurable

(D Gatzouras, 1999).

Hyperbolic Measures

Let X be a complete metric space and assume that

f : X !X is continuous. Let  be an f-invariant Borel

probability measure on X (i.e., (f

1

(A)) = (A)

for measurable sets A) with a compact support.

The Hausdorff dimension of , and the lower and

upper box dimensions of  (L-S Young, 1982) are

defined by

dim

 :¼ inffdim

Z : Z ˝ X;ðZÞ¼1g

dim

 :¼ lim

!0

inffdim

Z : Z ˝ X;ðZÞ1  g

dim

 :¼ lim

!0

inffdim

Z : Z ˝ X;ðZÞ1  g

It is natural to introduce the lower and upper

pointwise dimensions of  at x 2X by



ðxÞ:¼lim

r!0

log ð B

ðxÞÞ

log r

and similarly



(x). It has been shown by Young

(1982) that if X has finite topological dimension and

if  is exact dimensional, that is,



(x) = d



(x) =: d

for -a.e. x 2X, then

dim

 ¼ dim

 ¼ d

She also proved that hyperbolic measures (ergodic

measures with nonzero Lyapunov exponents), invar-

iant under a C

1þ

-diffeomorphism, >0, are exact

dimensional. F Ledrappier (1986) derived exact

dimensionality for hyperbolic Bowen–Ruelle–Sinai

measures. This result was extended by Ya Pesin and

Ch Yue (1996) to hyperbolic measures with semilocal

product structure. J-P Eckmann and D Ruelle (1985)

conjectured that the exact dimensionality holds for

general hyperbolic measures, and this was proved by

Barreira, Pesin, and Schmeling (1996). More precisely,

if f is a C

1þ

-diffeomorphism on a smooth Riemann

manifold X without boundary, and if  is f-invariant,

compactly supported Borel probability measure, then

its hyperbolicity implies that



ðxÞ¼d



ðxÞ¼d



ðxÞþd



ðxÞ

for -a.e. x 2X, where d



(x) and d



(x) are stable

and unstable pointwise dimensions of  at x

introduced by Ledrappier and Young (1985).

Multifractal Analysis of Functions and Measures

Invariant sets of many dynamical systems are not

self-similar. Roughly speaking, the aim of multi-

fractal analysis is to make a decomposition of the

invariant set with respect to desired fractal proper-

ties and then to study a fractal dimension of each

set of the decomposition. Some dynamical systems

have invariant sets equal to graphs of Ho¨ lderian

functions f : R

!R, so that wavelet methods can

be used. One of the goals of multifractal analysis

of functions is to study the spectrum of singularities

of f defined by

ðÞ:¼ dim



ðf Þ

introduced by U Frisch and G Parisi (1985) in the

context of fully developed turbulence. Here H



(f )is

the set of points at which the corresponding

pointwise Ho¨ lder exponent of f is equal to   0.

If the function f is self-similar then d

() is real

analytic and strictly concave (first increasing and

then decreasing) on an explicit interval (

a, a)

(S Jaffard, 1997). It is natural to consider the set

, 

(f ) of points x

called chirps of order (, )

(Y Meyer 1996), at which f behaves roughly

like jx  x



sin (1=jx  x



), >0. The function

(, ):= dim

, 

(f ) is called the chirp spectrum

of f (S Jaffard 2000). Wavelet methods have found

applications in the study of evolution equations

and in modeling and detection of chirps in

turbulent flows (S Jaffard, Y Meyer, R D Robert,

2001).

Basic ideas of multifractal analysis have been

introduced by physicists T Halsey, M H Jensen,

Fractal Dimensions in Dynamics 397

L P Kadanoff, I Procaccia, and B I Shraiman (1988).

In applications it often deals with an invariant

ergodic probability measure associated with the

dynamical system considered. Multifractal analysis

of a Borel finite measure  defined on R

consists in

the study of the function



ðÞ:¼ dim



ðÞ; 0

called the spectrum of pointwise dimensions of .

Here K



() is the set of points where the pointwise

dimension of  is equal to :



ðÞ: ¼fx 2R

: d



ðxÞ¼d



ðxÞ¼g

It is also of interest to study the Hausdorff

dimension of irregular set K():= {x 2R

: d



(x) <



(x)}. These sets are pairwise disjoint and consti-

tute a multifractal decomposition of R

, that is,

¼KðÞ[ [

 2R



ðÞðÞ

The function d



() provides an important informa-

tion about the complexity of multifractal decom-

position. In many situations, there is an open

interval (

, ) on which the function d



()is

analytic and strictly concave (first increasing and

then decreasing), and equal to the Legendre trans-

form of an explicit convex function. We thus obtain

an uncountable family of sets K



() with positive

Hausdorff dimension, which shows enormous com-

plexity of the multifractal decomposition of R

These and related questions have been studied by

L Olsen (1995), K Falconer (1996), Pesin and Weiss

(1996), Barreira and Schmeling (2000), and many

other authors.

Local Lyapunov Dimension

Let  be an open set in R

and let f :  !R

be a

-map. To any fixed x 2 we assign N singular

values a

 a

a

 0off, defined as square

roots of eigenvalues of the matrix f

(x)

 f

(x),

where f

(x) is the Jacobian of f at x, and f

(x)

its

transpose. The local Lyapunov dimension of f at x is

defined by

dim

ðf ; xÞ:¼j þ s

where j is the largest integer in [0, N]suchthat

a

 1 (if there is no such j we let j = 0), and

s 2[0, 1) is the unique solution of a

a

jþ1

= 1

(except for j = N, when we define s = 0). This

definition, due to B R Hunt (1996), is close to that of

Kaplan and Yorke (1979). The Jacobian f

(x)con-

tracts k-dimensional volumes (that is, a

a

< 1) if

and only if dim

(f , x) < k. In this case, we say that f is

k-contracting at x.Furthermore,thefunction

x 7! dim

(f , x) is upper-semicontinuous, so that for

any compact subset A of  the Lyapunov dimension

of f on A,

dim

ðf ; AÞ:¼ max

x 2A

dim

ðf ; xÞ

is well defined. Yu S Ilyashenko conjectured that if f

locally contracts k-dimensional volumes then the

upper box dimension of any compact invariant set is

<k. Hunt (1996) proved that if A is a compact,

strictly invariant set of f (i.e., f (A) = A) then

dim

A  dim

ðf ; AÞ½2

This is an improvement of dim

A  dim

(f , A)

obtained by A Douady and J Oesterle´ (1980), and

independently by Ilyashenko (1982). M A Blinchevs-

kaya and Yu S Ilyashenko (1999) proved that if A is any

attractor of a smooth map in a Hilbert space that

contracts k-dimensional volumes then

dim

A  k.See

[3] below.

A continuous variant of this method is used in

order to obtain estimates of fractal dimensions of

global attractors of dynamical systems (X, S)ona

Hilbert space X. Here S(t), t  0, is a semigroup of

continuous operators on X, that is, S(t þ s) = S(t)S(s)

and S(0) = I. A set A in X is called a global attractor

of dynamical system if it is compact, attracting

(i.e., for any bounded set B and ">0 there exists t

such that for t  t

we have S(t)B ˝ A

), and A is

strictly invariant (i.e., S(t)A = A for all t  0).

Applications in Dynamics

Logistic Map

M Feigenbaum, a mathematical physicist, intro-

duced and studied the dynamics of the logistic map



: [0, 1] ![0, 1], f



(x):= x(1  x),  2(0, 4]. Tak-

ing  = 

3.570 the corresponding invariant set

A [0, 1] (i.e., S

(A) [ S

(A) = A, where S

are two

branches of f

1



) has both Hausdorff and box

dimensions equal to 0.538 (P Grassberger 1981,

P Grassberger and I Procaccia, 1983). The set A

has Cantor-like structure, but is not self-similar.

Its multifractal properties have been studied by

U Frisch, K Khanin, and T Matsumoto (2004).

Smale Horseshoe

In the early 1960s S Smale defined his famous

horseshoe map and showed that it has a strange

invariant set resulting in chaotic dynamics. The

notion of strange attractor was introduced in 1971

by Ruelle and Takens in their study of turbulence.

Let S be a square in the plane and let f : R

a map transforming S as indicated in Figure 3, such

that on both components of S \ f

1

(S) the map f is

398 Fractal Dimensions in Dynamics

affine and preserves both horizontal and vertical

directions, and such that points 1, 2, 3, and 4 are

mapped to 1

, and 4

. Iterating f we get

backward invariant set 



:= \

j = 0

j

(S), forward

invariant set 

:= \

j = 0

(S), and invariant set

(horseshoe) 

:=

\ 



. These sets have the

Cantor set structure. More precisely, assuming that

the contraction parameter of f in vertical direction is

a 2(0, 1=2), and the expansion parameter in hor-

izontal direction is b > 2, then 

= [0, 1]  C

(a)

where C

(a)

is the Cantor set, 



= C

(1=b)

 [0, 1], and



= C

(1=b)

 C

(a)

, so that dim



= dim



= 1 þ

( log 2)=( log (1=a)) and

dim



¼ dim



log 2

log b

log 2

logð1=aÞ

This is a special case of a general result about

horseshoes in R

(not necessarily affine), due to

McCluskey and Manning (1983), stated in terms of

the pressure function. Analogous result as above can

be obtained for Smale solenoids. In R

it is possible

to construct affine horseshoes 

such that dim



dim



(M Pollicott and H Weiss, 1994).

Smale discovered a connection between homoclinic

orbits and the horseshoe map. It has been noticed that

fractal dimensions have important role in the study of

homoclinic bifurcations of nonconservative dynamical

systems. Since the 1970s the relationship between

invariants of hyperbolic sets and the typical dynamics

appearing in the unfolding of a homoclinic tangency

by a parametrized family of surface diffeomorphisms

has been studied by J Newhouse, J Palis, F Takens,

J-C Yoccoz, C G Moreira and M Viana. The main

result is that if the Hausdorff dimension of the

hyperbolic set involved in the tangency is <1then

the parameter set where the hyperbolicity prevails has

full Lebesgue density. If the Hausdorff dimension is

>1, then hyperbolicity is not prevalent. This result and

its proof were inspired by previous work of

J M Marstrand (1954) about arithmetic differences of

Cantor sets on the real line. According to the result by

Moreira, Palis, and Viana (2001) the paradigm

‘‘hyperbolicity prevails if and only if the Hausdorff

dimension is <1’’ extends to homoclinic bifurcations

in any dimension.

Using methods of thermodynamic formalism

McCluskey and Manning (1983) proved that if f

is the above horseshoe map, then there exists a

-neighborhood U of f such that the mapping

f 7! dim



is continuous. Continuity of box and

Hausdorff dimensions for horseshoes has been

studied also by Takens, Palis, and Viana (1988).

Lorenz Attractor

E N Lorenz (1963), a meteorologist and student of

G Birkhoff, showed by numerical experiments that

for certain values of positive parameters , r, b, the

quadratic system

x ¼ðy  xÞ;

y ¼rx  y  xz;

z ¼xy  bz

has the global attractor A, for example, for  = 10,

r = 28, b = 8=3. In this case dim

A 2.06, which is

a numerical result (Grassberger and Procaccia,

1983). Using the analysis of local Lyapunov dimen-

sion along the flow in A, G A Leonov (2001) showed

that if  þ 1  b  2 and r

(4  b) þ 2(b  1)

(2  3b) > b(b  1)

then

dim

A  3 

2ð þ b þ 1Þ

 þ 1 þ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ð  1Þ

þ 4r

He´ non Attractor

MHe´non (1976), a theoretical astronomer, discovered

the map f : R

, f (x, y):= (a þ by  x

, x), cap-

turing several essential properties of the Lorenz

system. In the case of a = 1.4 and b = 0.3, Hunt

(1996) derived from [2] that for any compact, strictly

f-invariant set A in the trapping region [1.8, 1.8]

∩ f

–1

(S )

∩ f

–1

(S ) ∩ f

–2

(S )

–2

(S ) ∩ f

–1

(S ) ∩ S ∩ f (S )∩ f

(S )

(s)

(A)

f (B )

(S )

4′′

4′

3′

2′

1′

1′′

2′′

3′′

Figure 3 The Smale horseshoe.

Fractal Dimensions in Dynamics 399

there holds dim

A < 1.5. Numerical experiments

show that dim

A 1.28 (Grassberger, 1983). Assum-

ing a > 0, b 2(0, 1), and P



, x



) 2A,whereP



are

fixed points of f, Leonov (2001) obtained that

dim

A  1 þ

1  ln b= lnð

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ



þ b

 x



Here



:¼

b  1 

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðb  1Þ

þ 4a



The proof is based on the study of local Lyapunov

dimension of f and its iterates on A.

Embedology

The physical relevance of box dimensions in the

study of attractors is related to the problem of

finding the smallest possible dimension n sufficient

to ‘‘embed’’ an attractor into R

.IfA R

is a

compact set and if n > 2

dim

A, then almost every

map from R

into R

, in the sense of prevalence, is

one-to-one on A and, moreover, it is an embedding

on smooth manifolds contained in A (T Sauer,

J A Yorke, and M Casdagli, 1991). If A is a strange

attractor then the same is true for almost every

delay-coordinate map from R

to R

. This improves

an earlier result by H Whitney (1936) and F Takens

(Takens’ embedology, 1981). The above notion of

prevalence means the following: a property holds

almost everywhere in the sense of prevalence if it

holds on a subset S of the space V := C

, R

) for

which there exists a finite-dimensional subspace

E V (probe space) such that for each v 2V we

have that v þ e 2S for Lebesgue a.e. e 2E.

Julia and Mandelbrot Sets

M Shishikura (1998) proved that the boundary of the

Mandelbrot set M generated by f

(z):= z

þ c has the

Hausdorff dimension equal to 2, thus answering

positively to the conjecture by B Mandelbrot,

J Milnor, and other mathematicians. Also for Julia

sets there holds dim

J(f

) = 2 for generic c in M

(i.e., on the set of second Baire category). The proof is

based on the study of the bifurcation of parabolic

periodic points. Also, each baby Mandelbrot set sitting

inside of M has the boundary of Hausdorff dimension

2 (L Tan, 1998). Shishikura’s results hold for more

general functions f (z):= z

þ c,whered  2.

For Julia sets J(f

) generated by f

(z):= z

þ c

there holds d(c):= dim

J(f

) = 1 þjcj

=(4 log 2) þ

o(jcj

) for c !0. This and more general results

have been obtained by Ruelle (1982). He also

proved that the function d(c) when restricted to the

interval [0, 1) is real analytic in [0, 1=4) [ (1=4, 1).

Furthermore, it is left continuous at 1/4 (O Bodart

and M Zinsmeister, 1996), but not continuous

(A Douady, P Sentenac, and M Zinsmeister, 1997).

Discontinuity of this map is related to the phenom-

enon of parabolic implosion at c = 1=4. The deriva-

tive d

(1=4  c)

d(1=4)3=2

(G Havard and M Zinsmeister,

2000). Here d(1=4) 1.07, which is a numerical

result. Analysis of dimensions is based on methods

of thermodynamic formalism.

C McMullen (1998) showed that if  is an

irrational number of bounded type (i.e., its contin-

ued fractional expansion [a

, a

, ...] is such that the

sequence (a

) is bounded from above) and

f (z):= z

þ e

2i

z, then the Julia set J(f ) is porous.

In particular,

dim

J(f ) < 2. Y C Yin (2000) showed

that if all critical points in J(f ) of a rational map

f :

C !C are nonrecurrent (a point is nonrecurrent

if it is not contained in its !-limit set) then J(f )is

porous, hence

dim

J(f ) < 2. Urban

ski and Przytycki

(2001) described more general rational maps such

that

dim

J(f ) < 2.

Spiral Trajectories

A standard planar model where the Hopf–Takens

bifurcation occurs is

r = r(r

l1

i = 0

’ = 1,

where l 2N.If is a spiral tending to the limit

cycle r = a of multiplicity m (i.e., r = a is a zero of

order m of the right-hand side of the first equation

in the system) then dim

=2  1=m. Furthermore,

for m > 1 the spiral is Minkowski measurable

ubrinic

and Z

upanovic

, 2005). For m = 1 the

spiral is Minkowski nondegenerate with respect to

the gauge function h("):= "( log (1="))

1

Infinite-Dimensional Dynamical Systems

In many situations the dynamics of the global attractor

A of the flow corresponding to an auto-

nomous Navier–Stokes system is finite-dimensional

(Ladyzhenskaya, 1972). This means that there exists a

positive integer N such that any trajectory in A is

completely determined by its orthogonal projection

onto an N-dimensional subspace of a Hilbert space X.

The aim is to find estimates of box and Hausdorff

dimensions of the global attractor, in order to under-

stand some of the basic and challenging problems of

turbulence theory. If A is a subset of a Hilbert space X,

its Hausdorff dimension is defined analogously as for

A R

. The definition of the upper box dimension

can be extended from A R

dim

A :¼lim

"!0

log mðA;"Þ

logð1="Þ

½3

400 Fractal Dimensions in Dynamics

where m(A, ") is the minimal number of balls

sufficient to cover a given compact set A X. The

value of log m(A, ") is called "-entropy of A.

Foias¸ and Temam (1979), Ladyzhenskaya (1982),

A V Babin and M I Vishik (1982), Ruelle (1983),

and E Lieb (1984) were among the first who

obtained explicit upper bounds of Hausdorff and

box dimensions of attractors of infinite-dimensional

systems. For global attractors A associated with

some classes of two-dimensional Navier–Stokes

equations with nonhomogeneous boundary condi-

tions it can be shown that

dim

A  c

G þ c

3=2

where G is the Grashof number, Re is the Reynolds

number, and c

are positive constants (R M Brown,

P A Perry, and Z Shen, 2000). V V Chepyzhov and

A A Ilyin (2004) obtained that

dim

A 

(1=

ﬃﬃﬃ

)(

jj)

1=2

G for equations with homoge-

neous boundary conditions, where  R

is a

bounded domain, and 

is the first eigenvalue of

. In the case of periodic boundary conditions

Constantin, Foias¸ , and Temam (1988) proved that

dim

A  c

2=3

(1 þ log G)

1=3

, while for a special

class of external forces there holds dim

A  c

2=3

(V X Liu, 1993). Let us mention an open problem by

V I Arnol’d: is it true that the Hausdorff dimension

of any attracting set of the Navier–Stokes equation

on two-dimensional torus is growing with the

Reynolds number?

In their study of partial regularity of solutions

of three-dimensional Navier–Stokes equations,

L Caffarelli, R Kohn, and L Nirenberg (1982)

proved that the one-dimensional Hausdorff mea-

sure in space and time (defined by parabolic

cylinders) of the singular set of any ‘‘suitable’’

weak solution is equal to zero. A weak solution is

said to be singular at a point (x

, t

)ifitis

essentially unbounded in any of its neighborhoods.

Dimensions of attractors of many other classes of

partial differential equations (PDEs) have been

studied, like for reaction–diffusion systems, wave

equations with dissipation, complex Ginzburg–

Landau equations, etc. Related questions for non-

autonomous PDEs have been considered by V V

Chepyzhov and M I Vishik since 1992.

Probability

Important examples of trajectories appearing in

physics are provided by Brownian motions. Brow-

nian motions ! in R

, N  2, have paths !([0, 1]) of

Hausdorff dimension 2 with probability 1, and they

are almost surely Hausdorff degenerate, since

(!([0, 1])) = 0 for a.e. ! (S J Taylor, 1953).

Defining gauge functions h("):= "

log (1=")

log log log (1=") when N = 2, and h("):= "

log (1=")

when N  3, there holds H

(!([0, 1])) 2(0, 1) for

a.e. ! (D Ray, 1963, S J Taylor, 1964). If N = 1 then

a.e. ! has the box and Hausdorff dimensions of

the graph of !j

[0, 1]

equal to 3/2 (Taylor, 1953), and

for the gauge function h("):= "

3=2

log log (1=") the

corresponding generalized Hausdorff measure is

nondegenerate. In the case of N  2 we have the

uniform dimension doubling property (R Kaufman,

1969). This means that for a.e. Brownian motion !

there holds dim

!(A) = 2 dim

A for all subsets

A [0, 1). There are also results concerning almost

sure Hausdorff dimension of double, triple, and

multiple points of a Brownian motion and of more

general Le´vy stable processes.

Fractal dimensions also appear in the study of

stochastic differential equations, like

¼ X

ðx

Þdt þ

k¼1

ðx

Þd

ðtÞ; x

¼ x 2R

The stochastic flow (x

)

t0

in R

is driven by a

Brownian motion ((t))

t0

in R

. Let us assume that

, k = 0, ..., d,areC

-smooth T-periodic divergence-

free vector fields on R

. Then for almost every

realization of the Brownian motion ((t))

t0

,thesetof

initial points x generating the flow (x

)

t0

with linear

escape to infinity (i.e.,

lim

t !1

(jx

j=t) > 0) is dense

and of full Hausdorff dimension N (D Dolgopyat,

V Kaloshin, and L Koralov, 2002).

Other Directions

There are many other fractal dimensions important

for dynamics, like the Re´nyi spectrum for dimen-

sions, correlation dimension, information dimen-

sion, Hentschel–Procaccia spectrum for dimensions,

packing dimension, and effective fractal dimension.

Relations between dimension, entropy, Lyapunov

exponents, Gibbs measures, and multifractal rigidity

have been investigated by Pesin, Weiss, Barreira,

Schmeling, etc. Fractal dimensions are used to study

dynamics appearing in Kleinian groups (D Sullivan,

C J Bishop, P W Jones, C McMullen, B O Stratmann,

etc.), quasiconformal mappings and quasiconfor-

mal groups (F W Gehring, J Va¨ isa¨la, K Astala,

C J Bishop, P Tukia, J W Anderson, P Bonfert-Taylor,

E C Taylor, etc.), graph directed Markov systems

(R D Mauldin, M Urban

ski, etc.), random walks on

fractal graphs (J Kigami, A Telcs, etc.), billiards

(H Masur, Y Cheung, P Ba´lint, S Tabachnikov,

N Chernov, D Sza´sz, IP To´th, etc.), quantum

dynamics (J-M Barbaroux, J-M Combes, H

Schulz-Baldes, I Guarneri, etc.), quantum gravity

(M Aizenman, A Aharony, M E Cates, T A Witten,

G F Lawler, B Duplantier, etc.), harmonic analysis

(R S Strichartz, Z M Balogh, J T Tyson, etc.),

Fractal Dimensions in Dynamics 401

number theory (L Barreira, M Pollicott, H Weiss,

B Stratmann, B Saussol, etc.), Markov processes

(R M Blumenthal, R Getoor, S J Taylor, S Jaffard,

C Tricot, Y Peres, Y Xiao, etc.), and theoretical

computer science (B Ya Ryabko, L Staiger,

J H Lutz, E Mayordomo, etc.), and so on.

See also: Bifurcations of Periodic Orbits; Chaos and

Attractors; Dissipative Dynamical Systems of Infinite

Dimension; Dynamical Systems in Mathematical Physics:

An Illustration from Water Waves; Ergodic Theory;

Generic Properties of Dynamical Systems; Holomorphic

Dynamics; Homoclinic Phenomena; Hyperbolic

Dynamical Systems; Image Processing: Mathematics;

Lyapunov Exponents and Strange Attractors; Partial

Differential Equations: Some Examples; Polygonal

Billiards; Quantum Ergodicity and Mixing of

Eigenfunctions; Stochastic Differential Equations;

Synchronization of Chaos; Universality and

Renormalization; Wavelets: Applications; Wavelets:

Mathematical Theory.

Further Reading

Barreira L (2002) Hyperbolicity and recurrence in dynamical

systems: a survey of recent results. Resenhas 5(3): 171–230.

Chepyzhov VV and Vishik MI (2002) Attractors for Equations of

Mathematical Physics, Colloquium Publications, vol. 49.

Providence, RI: American Mathematical Society.

Chueshov ID (2002) Introduction to the Theory of Infinite-

Dimensional Dissipative systems, ACTA Scientific Publ.

House. Kharkiv.

Falconer K (1990) Fractal Geometry. Chichester: Wiley.

Falconer K (1997) Techniques in Fractal Geometry. Chichester:

Wiley.

Ladyzhenskaya OA (1991) Attractors for Semigroups of

Evolution Equations. Cambridge: Cambridge University

Press.

Lapidus M and van Frankenhuijsen M (eds.) (2004) Fractal

Geometry and Applications: A Jubilee of Benoıˆt Mandelbrot,

Proc. Sympos. Pure Math., vol. 72, Parts 1 and 2, Providence,

RI: American Mathematical Society.

Mandelbrot B and Frame M (2002) Fractals. In: Encyclopedia of

Physical Science and Technology, 3rd edn., vol. 6, pp. 185–207.

San Diego, CA: Academic Press.

Mauldin RD and Urban

ski M (2003) Graph Directed Markov

Systems: Geometry and Dynamics of Limit Sets, Cambridge

Tracts in Mathematics, 148. Cambridge: Cambridge Univer-

sity Press.

Palis J and Takens F (1993) Hyperbolicity & Sensitive Chaotic

Dynamics at Homoclinic Bifurcations, Cambridge Studies in

Advanced Mathematics, 35. Cambridge: Cambridge Univer-

sity Press.

Pesin Ya (1997) Dimension Theory in Dynamical Systems: Con-

temporary Views and Applications, Chicago Lecture Notes in

Mathematics. Chicago, IL: University of Chicago Press.

Schmeling J and Weiss H (2001) An overview of the dimension

theory of dynamical systems. In: Katok A, de la Llave R, Pesin Ya,

and Weiss H (eds.) Smooth Ergodic Theory and Its Applications,

(Seattle, 1999), Proceedings of Symposia in Pure Mathematics,

vol. 69, pp. 429–488. Providence, RI: American Mathematical

Society.

Tan L (ed.) (2000) The Mandelbrot Set, Theme and Variations.

London Mathematical Society Lecture Note Series, 274.

Cambridge: Cambridge University Press.

Temam R (1997) Infinite-Dimensional Dynamical Systems in

Mechanics and Physics, 2nd edn. Applied Mathematical

Sciences, 68. New York: Springer.

Zinsmeister M (2000) Thermodynamic Formalism and Holo-

morphic Dynamical Systems, SMF/AMS Texts and Monographs

2. Providence, RI: American Mathematical Society, Socie´te´

Mathe´matique de France, Paris.

Fractional Quantum Hall Effect

J K Jain, The Pennsylvania State University,

University Park, PA, USA

Introduction

Interacting particles sometimes collectively behave

in ways that take us by complete surprise. In a

superfluid

He atoms flow without viscosity,

and in a superconductor electrons flow without

resistance. Such behaviors announce emergent

structures and principles which have often found

applications in other areas. This article concerns

the surprising collective effects that occur when

electrons are confined in two dimensions and

subjected to a strong transverse magnetic field.

At low temperatures, the Hall resistance (defined

below) exhibits plateaus on which it is precisely

quantized at

½1

where h and e are fundamental constants and f is a

plateau-specific rational fraction. This phenomenon

is known as the ‘‘fractional quantum Hall effect’’

(FQHE), or, after its discoverers, the ‘‘Tsui–Stormer–

Gossard’’ (TSG) effect. The underlying state provides

a new paradigm for collective behavior in nature, and

is understood in terms of a new class of quasiparticles

known as ‘‘composite fermions,’’ which are topologi-

cal bound states of electrons and quantized vortices.

This article will outline the basics of the experimental

phenomenology and our theoretical understanding of

this effect.

402 Fractional Quantum Hall Effect

The Hall Effect

The Ohm’s law, I = V=R, tells us that the current

through a resistor is proportional to the applied

voltage. The local form of the law is

J ¼ E ½2

where  is the conductivity, and J = qv is the

current density for particles of charge q and density

 moving with a velocity v.

In 1879, E H Hall discovered that in the presence of

a crossed electric and magnetic fields (E and B),

the current flows in a direction ‘‘perpendicular’’ to the

plane containing the two fields. Alternatively, the

passage of current induces a voltage perpendicular to

the direction of the current flow. This is known as

the Hall effect (see Figure 1). The phenomenon has a

classical origin. A consequence of the Lorentz force

law of electrodynamics,

F ¼ q E þ

v  B



½3

which gives the force on a particle of charge q

moving with a velocity v, is that for crossed electric

and magnetic fields the particle drifts in the

direction E  B with a velocity v = cE=B. The

current densi ty is therefore given by J = qv, where

 is the (three-dimensional) density of particles. That

produces the Hall resistivity



qc

½4

The von Klitzing Effect

Molecular beam epitaxy allows controllable layer

by layer growth in which one type of semiconductor,

say GaAs, can be grown on top of another, say

1x

As, to produce an atomically sharp interface.

By appropriately doping such structures, electrons can

be captured at the interface, thus producing a two-

dimensional electron system (2DES). We note that

these are three-dimensional electrons confined to move

in two dimensions. The interaction has the standard

Coulomb form V(r) = e

=r, where  is the dielectric

constant of the host material. (In a hypothetical world

which has only two space dimensions, the interaction

would be logarithmic.)

The ‘‘integral quantum Hall effect’’ (IQHE) or the

‘‘von Klitzing effect’’ was discovered unexpectedly

by von Klitzing and collaborators in 1980, in their

study of Hall effect in a 2DES. In two dimensions,

one defines the Hall resistance as

½5

which, from classical electrodynamics, is expected to

be proportional to the magnetic field B.Thatisindeed

the case at small magnetic fields. At sufficiently high B,

however, quantum mechanical effects appear in a

dramatic manner. The essential observations are as

follows.

1. When plotted as a function of the magne tic field

B, the Hall resistance exhibits numerous plateaus. On

any given plateau, R

is precisely quantized with

values given by

½6

where n is an integer (hence the name ‘‘integral

quantum Hall effect’’). The plateau occurs in the

vicinity of   Be=hc = n, where  is the ‘‘filling

factor’’ (defined below).

2. In the plateau region, the longitudinal resis-

tance exhibits an Arrhenius behavior:

 exp 





½7

This gives a filling-factor dependent energy scale ,

which indicates the presence of a gap in the

excitation spectrum. R

vanishes in the limit T ! 0.

The absolute accuracy of the quantization has

been established to a few parts in 10

for 1

uncertainty, and the relative accuracy to a few

parts in 10

. There is presently no known

‘‘intrinsic’’ c orrection t o the quantization. Perhaps,

the most remarkable aspect of t he effect is its

universality. It is indepe ndent of the sample type,

geometry, various material parameters (the band

mass of the electron or the dielectric constant of the

semiconductor), and disorder. The combination

h=e

also occurs in the defini tion of the fine

Figure 1 Schematics of magnetotransport measurement. I, V

and V

are the current, longitudinal voltage, and the Hall

voltage, respectively. The longitudinal and Hall resistances are

defined as R

V

=I and R

 V

=I.

Fractional Quantum Hall Effect 403

structure constant  = e

=hc, the value of which is

approximately 1/137. The Hall effect measure-

ments in dirty, solid state systems thus provide

one of the most accurate values for . Finally, the

lack of resist ance at T = 0 is to be c ontrasted with

ordinary metals, for which the resistance at T ! 0,

called the residual resistance, is finite and propor-

tional to disorder.

The TSG Effect

The next revolution occurred in 1982 with the

discovery of the TSG effect, that is, plat eaus on

which the Hall resistance is quantized at values

given by eqn [1] (see Figure 2). The observation

of the R

= h=fe

plateau is often referred to as

the observation of the fraction f. Improvement

of experimental conditions has led to the observa-

tion of a large number of fractions over the

years, revealin g the richness of the TSG effect.

At the time of the writing of this article, the

number of observed fractions is more than 50 if

one counts only fractions below unity. As in the

von Klitzing effect, the longitudinal resistance

exhibits an Arrhenius behavior, vanishing in the

limit T ! 0.

Landau Levels

The Hamiltonian for a nonrelativistic electron

moving in tw o space dimensions in a perpendicular

magnetic field is given by

H ¼

p þ



½8

Here, m

is the electron’s band mass and e its

charge. For a uniform magnetic field, the vector

potential A satisfies

Ñ  A ¼ B

z ½9

Because A is a linear function of the spatial

coordinates, it follows that H is a generalized two-

dimensional harmonic oscillator Hamiltonian which

is quadratic in both the spatial coordinates and in

the canonical momentum p = ihÑ, and therefore

can be diagonalized exactly.

A convenient gauge choice is the symmetric gauge:

A ¼

B  r

ðy; x; 0Þ½10

With the magnetic length ‘ =

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

hc=eB

and the

cyclotron energy h!

=heB =m

c chosen as the

2.5

1.5

0.5

(h /e

)

4/3

5/7

3/7

4/7

4/5

1/3

2/5

10 20 30

4/9

5/9

2/3

3/5

7/5

5/3

netic field (T)

43 2

Figure 2 The TSG effect. The Hall resistance (R

) exhibits many precisely quantized plateaus, concurrent with minima in the

longitudinal resistance (R). Reproduced with permission from Perspectives in Quantum Hall Effects; HL Stormer and DC Tsui;

SD Sarma and A Pinczuk (eds.); Copyright ª 1997, Wiley. Reprinted with permission of John Wiley & Sons, Inc.

404 Fractional Quantum Hall Effect

units for length and energy , the Hamiltonian can be

expressed as

H ¼

i





þi



½11

Choosing as independent variables

z  x  iy;



z  x þ iy ½12

we get

H ¼

4

@z@



z  z





½13

Now define the following sets of ladder operators:

b ¼

ﬃﬃﬃ



þ 2



½14

ﬃﬃﬃ

 2





½15

ﬃﬃﬃ



 2



½16

a ¼

ﬃﬃﬃ

þ 2





½17

which have the property that

½a; a

¼1; ½b; b

¼1 ½18

and all the other commutators are zero. In terms of

these operators, the Hamiltonian can be written as

H ¼ a

a þ

½19

The eigenvalue of a

a is an integer, n, called the

Landau level (LL) index. The z-component of the

canonical angular momentum operator, the only

relevant component for the two-dimensional prob-

lem, is defined as

¼i

@



 z

¼ a

a  b

b ½20

Exploiting the property [H, L

] = 0, the eigenfunctions

will be chosen to diagonalize H and L

simultaneously.

The eigenvalue of L

will be denoted by m.The

analogy to the Harmonic oscillator problem immedi-

ately gives the solution

Hjm; ni¼E

jm; ni½21

where

¼ n þ



½22

and

jm; ni¼

ðb

mþn

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðm þ nÞ!

ða

ﬃﬃﬃﬃ

j0; 0i½23

where m = n, n þ 1, .... The single-particle orbital

at the bottom of the two ladders defined by the two sets

of raising and lowering operators is

hrj0; 0i

0;0

ðrÞ¼

ﬃﬃﬃﬃﬃﬃ

2

z



z=4

½24

which satisfies

aj0; 0i¼bj0; 0i¼0 ½25

The single-p article states are particularly simple in

the lowest Landau level (n = 0):



0;m

ðrÞ¼hrj0; mi¼

z



z=4‘

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2‘

½26

Aside from the ubiquitous Gaussian factor, a general

state in the lowest Landau level is given by a

polynomial of z; it does not involve any



z.Inother

words, apart from the Gaussian factor, the lowest

Landau level wave functions are analytic functions of z.

Landau Level Degeneracy

The state 

0, m

(r) is peaked strongly at r =

ﬃﬃﬃﬃﬃﬃﬃ

‘.

Neglecting order-1 effects, there are m states in the

lowest Landau level in a disk of radius r =

ﬃﬃﬃﬃﬃﬃﬃ

‘,

giving a degeneracy of (2‘

)

1

per unit area per

Landau level. (The same degeneracy is obtained for

higher Landau levels as well.) It is equal to B=

where 

= hc=e is called the flux quantum, that is,

there is one state per flux quantum in each Landau

level.

Filling Factor

The number of filled Landau levels, called the filling

factor, is given by

 ¼2‘



½27

The Origin of Plateaus

The von Klitzing effect can be explained in terms of

a model which neglects the interactions between

electrons. It occurs because the ground state at an

integral filling is unique and nondegenerate, sepa-

rated from excitations by a gap. Laughlin (1981)

showed that the disorder-induced Anderson locali-

zation also plays a crucial role in the establishment

of the Hall plateaus. To see this, imagine changing

the filling away from an integer by adding some

Fractional Quantum Hall Effect 405