Fractal Dimensions in Dynamics

Fractal Dimensions in Dynamics
MAPH 00373
Fractal Dimensions in Dynamics
V Županović and D Žubrinić, University of Zagreb,
Zagreb, Croatia
s-dimensional Minkowski content of A, s 0,
we mean
ª 2006 Elsevier Ltd. All rights reserved.
Ms ðAÞ :¼ lim
dimB A :¼ inffs 0: Ms ðAÞ ¼ 0g
The lower s-dimensional Minkowski content Ms (A)
and the corresponding lower box dimension dimB A
are defined analogously. The name of box dimension stems from the following: if we have an "-grid
in RN composed of closed N-dimensional boxes
with side ", and if N(A, ") is the number of boxes of
the grid intersecting A, then
Since the 1970s, dimension theory for dynamics has
evolved into an independent field of mathematics.
Its main goal is to measure complexity of invariant
sets and measures using fractal dimensions. The
history of fractal dimensions is closely related to
the names of H Minkowski (Minkowski content,
1903), H Hausdorff (Hausdorff dimension,
1919), G Bouligand (Bouligand dimension, 1928),
LS Pontryagin and LG Schnirelmann (metric order,
1932), P Moran (Moran geometric constructions,
1946), AS Besicovitch and SJ Taylor (Besicovitch–
Taylor index, 1954), A Rényi (Rényi spectrum
for dimensions, 1957), AN Kolmogorov and
VM Tihomirov (metric dimension, Kolmogorov
complexity, 1959), YaG Sinai, D Ruelle, R Bowen
(thermodynamic formalism, Bowen’s equation,
1972, 1973, 1979), B Mandelbrot (fractals and
multifractals, 1974), JL Kaplan and JA Yorke
(Lyapunov dimension, 1979), JE Hutchinson (fractals and self-similarity, 1981), C Tricot, D Sullivan
(packing dimension, 1982, 1984), HGE Hentschel
and I Procaccia (Hentschel–Procaccia spectrum for
dimensions, 1983), Ya Pesin (Carathéodory–Pesin
dimension, 1988), M Lapidus and M van Frankenhuysen (complex dimensions for fractal strings,
2000), etc. Fractal dimensions enable us to have a
better insight into the dynamics appearing in various
problems in physics, engineering, chemistry, medicine, geology, meteorology, ecology, economics,
computer science, image processing, and, of course,
in many branches of mathematics. Concentrating on
box and Hausdorff dimensions only, we describe
basic methods of fractal analysis in dynamics, sketch
their applications, and indicate some trends in this
rapidly growing field.
dimB A ¼ lim
Fractal Dimensions
Box Dimensions
Let A be a bounded set in R N , and let d(x, A) be
Euclidean distance from x to A. The Minkowski
sausage of radius " around A (a term coined by
B Mandelbrot) is defined as "-neighborhood of A,
that is, A" := {y 2 RN: d(y, A) < "}. By the upper
log NðA; "Þ
and analogously for dimB A. It suffices to take any
geometric subsequence "k = bk in the limit, where
b > 1 (H. Furstenberg, 1970). There are many other AU3
names for the upper box dimension appearing in the
literature, like the Cantor–Minkowski order, Minkowski dimension, Bouligand dimension, Borel
logarithmic rarefaction, Besicovitch–Taylor index,
entropy dimension, Kolmogorov dimension, fractal
dimension, capacity dimension, and limit capacity.
If A is such that dimB A = dimB A, the common value
is denoted by d := dimB A, and we call it the box
dimension of A. If, in addition to this, both Md (A)
and Md (A) are in (0, 1), we say that A is
Minkowski nondegenerate. If, moreover, Md (A) =
Md (A) =: Md (A) 2 (0, 1), then A is said to be
Minkowski measurable.
Assume that A is such that d := dimB A and p0015
Md (A) exist. Then the value of Md (A)1 is called
the lacunarity of A (B Mandelbrot, 1982). A AU4
bounded set A R N is said to be porous (A Denjoy,
1920) if there exist > 0 and > 0 such that
for every x 2 A and r 2 (0, ) there is y 2 R N such
that the open ball Br (y) is contained in Br (x) n A.
If A is porous then it is easy to see that dimB A < N
(O Martio and M Vuorinen, 1987, A Salli, 1991).
Let p0020
A := C(a) , a 2 (0, 1=2), be the Cantor set obtained
from [0, 1] by consecutive deletion of 2k middle
open intervals of length ak (1 2a) in step k 2 N [
{0}. Then dimB A = ( log 2)=( log (1=a)) (G Bouligand,
1928), and A is nondegenerate, but not Minkowski
measurable (Lapidus and Pomerance, 1993). For
the spiral of focus type defined by r = m’ in
polar coordinates, where 2 (0, 1) and m > 0 are
Here jj denotes N-dimensional Lebesgue measure.
The corresponding upper box dimension is defined by
AU2 s0005
jA" j
2 ½0; 1
MAPH 00373
2 Fractal Dimensions in Dynamics
Hausdorff Dimension
For a given subset A of RN (not necessarily
s 0 we define Hs (A) := lim" ! 0
P1 and
inf { i = 1 ri } 2 [0, 1], where the infimum is taken
over all finite or countable coverings of A by open
balls of radii ri ". The value of Hs (A) is called
s-dimensional Hausdorff outer measure of A. The
Hausdorff dimension of A, sometimes called the
Hausdorff–Besicovitch dimension, is defined by
If A is bounded then dimH A dimB A dimB A N.
We say that A is Hausdorff nondegenerate (or d-set)
if Hd (A) 2 (0, 1) for some d 0. Cantor sets share
this property, and dimH C(a) = ( log 2)=( log (1=a)),
where a 2 (0, 1=2) (Hausdorff, 1919).
Gauge Functions
The notions of Minkowski contents and Hausdorff
measure can be generalized using gauge functions
h : [0, "0 ) ! R that are assumed to be continuous,
increasing, and h(0) = 0. For example,
MhðAÞ :¼ lim
jA" j
and similarly for Mh (A) (M Lapidus and C He,
1997), while for Hh (A) it suffices to change ri s with
h(ri ) in the above definition of the Hausdorff outer
measure (Besicovitch, 1934). Gauge functions are
used for sets that are Minkowski or Hausdorff
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
fj 2 Q: j½n¼i½ng k¼0
The topological entropy of j Q is defined by
h(jQ) := P(0), that is,
dimH A :¼ inffs 0: Hs ðAÞ ¼ 0g
Thermodynamic formalism has been developed by p0040
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 Xp be the set of all sequences i = (i1 , i2 , . . . ) of
elements ik from a given set of p symbols, say
. . . , p}. We endow Xp with the metric d(i, j) :=
P 2, k
k 2 jik jk j and introduce the one-sided shift
operator (or left shift) : Xp ! Xp defined by
((i))n = inþ1 , that is, (i1 , i2 , i3 , . . . ) = (i2 , i3 , i4 , . . . ).
A set Q ˝ Xp 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] := (i1 , . . . , in ). Given a continuous function
’ : Q ! R, let us define the topological pressure of
’ with respect to by
Pð’Þ :¼ lim log
n!1 n
fi½n:i 2 Qg
Eði½nÞ :¼ exp
’ð ðjÞÞ
fixed, ’ ’1 > 0, we have dimB = 2=(1 þ )
(M Mendés-France, Y Dupain, C Tricot, 1983). It
is Minkowski measurable (Žubrinić and Županović,
2005), and the larger m, the smaller the lacunarity;
see Figure 1.
Thermodynamic Formalism
Figure 1 Spirals of equal box dimensions (4/3) and different
lacunarities (0.43 and 0.05).
Methods of Fractal Analysis in Dynamics
hðjQÞ ¼ lim
n!1 n
log #fi½n: i 2 Qg
where # denotes the cardinal
number of a set. The
above function ’n := n1
k has the property
’nþm = ’n þ ’m , and therefore we speak about
additive thermodynamic formalism. Topological pressure 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 2 R, and with a suitably chosen function ’, this
equation reads
Pðs’Þ ¼ 0
Geometric Constructions
A geometric construction (Q, ) in Rm indexed by p0045
symbolic dynamics Q is a family of compact sets
i[n] Rm , i 2 Q, n 2 N, such that diami[n] ! 0 as
n ! 1, i[nþ1] ˝ i[n] , i[n] = inti[n] for every i 2 Q
and all n, and inti[n] \ intj[n] = ; whenever
i[n] 6¼ j[n] (Moran’s open set condition). This family
induces the Cantor-like set
MAPH 00373
Fractal Dimensions in Dynamics
Figure 2 Cantor-like set.
F :¼
(see Figure 2). The mapping h : Q ! F defined by
h(i) := \1
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 1 , . . . , p are given numbers in (0, 1), and i[n]
are balls of radii ri[n] := i1 . . . in , then s := dimH F is
the unique solution of Bowen’s equation P(s’) = 0,
where ’ is defined by ’(i) := log i1 (Ya Pesin and
H Weiss, 1996). In this case Bowen’s equation is
equivalent to Moran’s equation (1946),
k s ¼ 1
Hyperbolic Measures
Let X be a complete metric space and assume that p0060
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
It is a compact, -invariant subset of Xp . The map
j XA is called the subshift of finite type (Bowen,
1975). A construction of Cantor-like set F using
dynamics Q = Xp is called a simple geometric construction, while a geometric construction is said to be a
Markov geometric construction if Q = XA . If F is
obtained by a Markov geometric construction such
that all i[n] are balls of radii ri[n] := i1 . . . in , where
ij 2 (0, 1), ij 2 {1, . . . p}, then dimB F = dimH F = s,
where s is the unique solution of equation
(AMs ) = 1. Here Ms := diag(1 s , . . . , p s ) and
(AMs ) is the spectral radius of the matrix AMs . This
and more general results have been obtained by Pesin
and Weiss (1996).
Any Cantor-like set F obtained via iterated p0055
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 1 , . . . , log p } is not contained in r Z
for any r > 0, then F is Minkowski measurable
(D Gatzouras, 1999).
This result has been generalized by L Barreira (1996)
using the Carathéodory–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 ri[nþ1] ri[n] and ri[nþm] ri[n] rn (i)[m] for
all i 2 Q, n, m 2 N. Then dimH F = dimB F = s, where
s is the unique real number such that
ri½n s ¼ 0
lim log
n!1 n
fi½n:i 2 Qg
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
ri[n] := i1 . . . in , where i = (i1 , i2 , . . . ), and 1 , . . . , p 2
(0, 1) are given numbers. Pesin and Weiss (1996)
showed that Moran’s open set condition 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 (Ruelle, 1982),
horseshoes (H McCluskey and A Manning, 1983), etc.
An important example of symbolic dynamics is the
topological Markov chain XA generated by a p p
matrix A with entries aij 2 {0, 1}:
XA :¼ fi ¼ ði1 ; i2 ; . . .Þ 2 Xp: aik ikþ1 ¼ 1 for all k 2 Ng
dimH :¼ inffdimH Z : Z ˝ X; ðZÞ ¼ 1g
dimB :¼ lim inffdimB Z : Z ˝ X; ðZÞ 1 g
dimB :¼ lim inffdimB Z : Z ˝ X; ðZÞ 1 g
It is natural to introduce the lower and upper
pointwise dimensions of at x 2 X by
log ðBr ðxÞÞ
log r
d ðxÞ :¼ lim
and similarly d (x). It has been shown by Young
(1982) that if X has finite topological dimension and
if is exact dimensional, that is, d (x) = d (x) =: d
for -a.e. x 2 X, then
dimH ¼ dimB ¼ d
She also proved that hyperbolic measures (ergodic
measures with nonzero Lyapunov exponents), invariant under a C1þ -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)
MAPH 00373
4 Fractal Dimensions in Dynamics
Multifractal Analysis of Functions and
Invariant sets of many dynamical systems are not
self-similar. Roughly speaking, the aim of multifractal analysis is to make a decomposition of the
invariant set with respect to desired fractal properties and then to study a fractal dimension of each
set of the decomposition. Some dynamical systems
have invariant sets equal to graphs of Hölderian
functions f : R N ! 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
K ðÞ :¼ fx 2 RN: d ðxÞ ¼ d ðxÞ ¼ g
It is also of interest to study the Hausdorff
dimension of irregular set K() := {x 2 RN: d (x) <
d (x)}. These sets are pairwise disjoint and constitute a multifractal decomposition of RN , that is,
RN ¼ KðÞ [ ð[ 2 R K ðÞÞ
The function d () provides an important information about the complexity of multifractal decomposition. 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 transform of an explicit convex function. We thus obtain
an uncountable family of sets K () with positive
Hausdorff dimension, which shows enormous complexity of the multifractal decomposition of RN .
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.
called the spectrum of pointwise dimensions of .
Here K () is the set of points where the pointwise
dimension of is equal to :
for -a.e. x 2 X, where ds (x) and du (x) are stable
and unstable pointwise dimensions of at x
introduced by Ledrappier and Young (1985).
d ðxÞ ¼ d ðxÞ ¼ ds ðxÞ þ du ðxÞ
d ðÞ :¼ dimH K ðÞ;
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 C1þ -diffeomorphism on a smooth Riemann
manifold X without boundary, and if is f-invariant,
compactly supported Borel probability measure, then
its hyperbolicity implies that
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 Hölder exponent of f is equal to 0.
If the function f is self-similar then df () 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
C, (f ) of points x0 called chirps of order (, )
(Y Meyer 1996), at which f behaves roughly like
jx x0 j sin (1=jx x0 j ), > 0.
Df (, ) := dimH C, (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, RD Robert 2001).
Basic ideas of multifractal analysis have been
introduced by physicists T Halsey, MH Jensen,
LP Kadanoff, I Procaccia, and BI 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 RN consists in
the study of the function
Local Lyapunov Dimension
df ðÞ :¼ dimH H ðf Þ
Let be an open set in RN and let f : ! RN be a p0075
C1 -map. To any fixed x 2 we assign N singular
values a1 a2 aN 0 of f, defined as square
roots of eigenvalues of the matrix f 0 (x)> f 0 (x),
where f 0 (x) is the Jacobian of f at x, and f 0 (x)> its
transpose. The local Lyapunov dimension of f at x is
defined by
dimL ðf ; xÞ :¼ j þ s
where j is the largest integer in [0, N] such that
a1 aj 1 (if there is no such j we let j = 0), and
s 2 [0, 1) is the unique solution of a1 aj asjþ1 = 1
(except for j = N, when we define s = 0). This
definition, due to BR Hunt (1996), is close to that of
Kaplan and Yorke (1979). The Jacobian f 0 (x) contracts k-dimensional volumes (that is, a1 ak < 1) if
and only if dimL (f , x) < k. In this case, we say that f is
k-contracting at x. Furthermore, the function
x 7! dimL (f , x) is upper-semicontinuous, so that for
any compact subset A of the Lyapunov dimension
of f on A,
dimL ðf ; AÞ :¼ max dimL ðf ; xÞ
is well defined. Yu S Ilyashenko conjectured that if f
locally contracts k-dimensional volumes then the
MAPH 00373
Fractal Dimensions in Dynamics
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
This is an improvement of dimH A dimL(f , A)
obtained by A Douady and J Oesterlé (1980), and
independently by Ilyashenko (1982). MA Blinchevskaya
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 dimB 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) on a
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 we have
S(t)B ˝ A" ), and A is strictly invariant (i.e., S(t)A = A
for all t 0).
S ∩ f –1(S )
f (s)
dimB A dimL ðf ; AÞ
f (A)
f (B )
S ∩ f –1(S ) ∩ f –2(S )
f –2(S ) ∩ f –1(S ) ∩ S ∩ f (S )∩ f 2(S )
Applications in Dynamics
Logistic Map
M Feigenbaum, a mathematical physicist, introduced and studied the dynamics of the logistic map
f : [0, 1] ! [0, 1], f (x) := x(1 x), 2 (0, 4]. Taking = 1 3.570 the corresponding invariant set
A [0, 1] (i.e., S1 (A) [ S2 (A) = A, where Si are two
branches of f1 ) 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 : R2 ! R2 be
a map transforming S as indicated in Figure 3, such
that on both components of S \ f 1 (S) the map f is
affine and preserves both horizontal and vertical
directions, and such that points 1, 2, 3, and 4 are
mapped to 10 , 20 , 30 , and 40 . Iterating f we get
backward invariant set := \1
j = 0 f (S), forward
invariant set þ := \j = 0 f (S), and invariant set
(horseshoe) f := þ \ . These sets have the
Cantor set structure. More precisely, assuming that
f 2(S )
Figure 3 The Smale horseshoe.
the contraction parameter of f in vertical direction is
a 2 (0, 1=2), and the expansion parameter in horizontal direction is b > 2, then þ = [0, 1] C(a) ,
where C(a) is the Cantor set, = C(1=b) [0, 1], and
f = C(1=b) C(a) , so that dimB þ = dimH þ = 1 þ
( log 2)=( log (1=a)) and
dimB f ¼ dimH f ¼
log 2
log 2
log b logð1=aÞ
This is a special case of a general result about
horseshoes in R 2 (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 R3 it is possible
to construct affine horseshoes f such that dimH f <
dimB f (M Pollicott and H Weiss, 1994).
Smale discovered a connection between homoclinic p0095
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
MAPH 00373
6 Fractal Dimensions in Dynamics
dimB A 1 þ
1 ln b= lnð x2 þ b x Þ
x :¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
b 1 ðb 1Þ2 þ 4a
The proof is based on the study of local Lyapunov
dimension of f and its iterates on A.
The physical relevance of box dimensions in the p0115
study of attractors is related to the problem of
finding the smallest possible dimension n sufficient
to ‘‘embed’’ an attractor into R n . If A Rk is a
compact set and if n > 2dimB A, then almost every
map from Rk into Rn , 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,
JA Yorke, and M Casdagli, 1991). If A is a strange
attractor then the same is true for almost every
delay-coordinate map from Rk to Rn . 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 := C1 (R k , R n ) for
which there exists a finite-dimensional subspace
E V (probe space) such that for each v 2 V we
have that v þ e 2 S for Lebesgue a.e. e 2 E.
has been studied by J Newhouse, J Palis, F Takens,
J-C Yoccoz, CG Moreira and M Viana. The main
result is that if the Hausdorff dimension of the
hyperbolic set involved in the tangency is < 1 then
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
JM 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
C1 -neighborhood U of f such that the mapping
f 7! dimH f is continuous. Continuity of box and
Hausdorff dimensions for horseshoes has been
studied also by Takens, Palis, and Viana (1988).
Lorenz Attractor
EN 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_ ¼ rx y xz;
z_ ¼ xy bz
has the global attractor A, for example, for = 10,
r = 28, b = 8=3. In this case dimB A 2.06, which is
a numerical result (Grassberger and Procaccia,
1983). Using the analysis of local Lyapunov dimension along the flow in A, GA Leonov (2001) showed
that if þ 1 b 2 and r2 (4 b) þ 2(b 1)
(2 3b) > b(b 1)2 then
2ð þ b þ 1Þ
þ 1 þ ð 1Þ2 þ 4r
dimB A 3 Hénon Attractor
M Hénon (1976), a theoretical astronomer, discovered the map f : R2 ! R2 , f (x, y) := (a þ by x2 , x),
capturing 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]2
there holds dimB A < 1.5. Numerical experiments
show that dimB A 1.28 (Grassberger, 1983).
Assuming a > 0, b 2 (0, 1), and P (x , x ) 2 A,
where P are fixed points of f, Leonov (2001)
obtained that
Julia and Mandelbrot Sets
M Shishikura (1998) proved that the boundary of the p0120
Mandelbrot set M generated by fc (z) := z2 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 dimH J(fc ) = 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) := zd þ c,
where d 2.
For Julia sets J(fc ) generated by fc (z) := z2 þ c p0125
there holds d(c) := dimH J(fc ) = 1 þ jcj2 =(4 log 2) þ
o(jcj2 ) 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).
MAPH 00373
Fractal Dimensions in Dynamics
A standard planar model where P
the Hopf–Takens
bifurcation occurs is r_ = r(r2l þ l1
˙ = 1,
i = 0 ai r ), ’
where l 2 N. 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 dimB = 2 1=m. Furthermore,
for m > 1 the spiral is Minkowski measurable
(Žubrinić and Županović, 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 autonomous 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 understand 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 RN . The definition of the upper box dimension
can be extended from A RN to
Spiral Trajectories
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 conditions it can be
shown that dimB A c1 G þ c2 Re3=2 , where G is the
Grashof number, Re is the Reynolds number, and ci
are positive constants (RM Brown, PA Perry, and
Z Shen, 2000). VV Chepyzhovpand
ffiffiffi AA Ilyin (2004)
obtained that dimB A (1= 2
)(1 jj)1=2 G for
equations with homogeneous boundary conditions,
where R 2 is a bounded domain, and 1 is the
first eigenvalue of . In the case of periodic
boundary conditions Constantin, Foiaş, and
Temam (1988) proved that dimB A c1 G2=3 (1 þ
log G)1=3 , while for a special class of external forces
there holds dimH A c2 G2=3 (VX Liu, 1993). Let us
mention an open problem by VI 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 p0150
of three-dimensional Navier–Stokes equations,
L Caffarelli, R Kohn, and L Nirenberg (1982)
proved that the one-dimensional Hausdorff measure
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 (x0 , t0 ) if it is 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 nonautonomous PDEs have been considered by
VV Chepyzhov and MI Vishik since 1992.
Discontinuity of this map is related to the phenomenon of parabolic implosion at c = 1=4. The derivative d0 (c) tends to þ1 from the left at c = 1=4 like
(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 continued fractional
expansion [a1 , a2 , . . . ] is such that the sequence (ai ) is
bounded from above) and f (z) := z2 þ e2
i z, then the
Julia set J(f ) is porous. In particular, dimB J(f ) < 2.
YC 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 dimB J(f ) < 2. Urbański and
Przytycki (2001) described more general rational maps
such that dimB J(f ) < 2.
dimB A :¼ lim"!0
log mðA; "Þ
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.
Foiaş and Temam (1979), Ladyzhenskaya (1982),
AV Babin and MI Vishik (1982), Ruelle (1983), and
Important examples of trajectories appearing in p0155
physics are provided by Brownian motions. Brownian motions ! in RN , N 2, have paths !([0, 1]) of
Hausdorff dimension 2 with probability 1, and they
are almost surely Hausdorff degenerate, since
H2 (!([0, 1])) = 0 for a.e. ! (SJ Taylor, 1953).
Defining gauge functions h(") := "2 log (1=")
log log log (1=") when N = 2, and h(") := "2 log (1=")
when N 3, there holds Hh (!([0, 1])) 2 (0, 1) for
a.e. ! (D Ray, 1963, SJ 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
MAPH 00373
8 Fractal Dimensions in Dynamics
Xk ðxt Þ dk ðtÞ;
x0 ¼ x 2 RN
Further Reading
Barreira L (2002) Hyperbolicity and recurrence in dynamical
systems: a survey of recent results, arXiv:math.DS/0210267.
Chepyzhov VV and Vishik MI (2002) Attractors for Equations of
Mathematical Physics, Colloquium Publications vol. 49.
American Mathematical Society.
Chueshov ID (2002) Introduction to the Theory of InfiniteDimensional Dissipative systems, ACTA Scientific Publ.
House. Kharkiv.
Falconer K (1990) Fractal Geometry. Wiley.
Falconer K (1997) Techniques in Fractal Geometry. Wiley.
Ladyzhenskaya OA (1991) Attractors for Semigroups of Evolution Equations. Cambridge University Press.
Lapidus M and van Frankenhuijsen M (eds.) (2004) Fractal
Geometry and Applications: A Jubilee of Benoı̂t 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.
Academic Press.
Mauldin RD and Urbański M (2003) Graph Directed Markov
Systems: Geometry and Dynamics of Limit Sets, Cambridge
Tracts in Mathematics.
Palis J and Takens F (1993) Hyperbolicity & Sensitive Chaotic
Dynamics at Homoclinic Bifurcations. Cambridge University Press.
Pesin Ya (1997) Dimension Theory in Dynamical Systems:
Contemporary Views and Applications, Chicago Lecture
Notes in Mathematics Series.
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. American Mathematical Society.
Tan L (ed.) (2000) The Mandelbrot Set, Theme and Variations.
Cambridge University Press.
Temam R (1997) Infinite-Dimensional Dynamical Systems in
Mechanics and Physics, 2nd edn. Springer.
Zinsmeister M (2000) Thermodynamic Formalism and Holomorphic Dynamical Systems, SMF/AMS Texts and Monographs
2. Providence, RI: American Mathematical Society, Société
Mathématique de France, Paris.
The stochastic flow (xt )t0 in RN is driven by a
Brownian motion ((t))t0 in R d . Let us assume that
Xk , k = 0, . . . , d, are C1 -smooth T-periodic divergencefree vector fields on RN . Then for almost every
realization of the Brownian motion ((t))t0 , the set of
initial points x generating the flow (xt )t0 with linear
escape to infinity (i.e., limt ! 1 (jxt j=t) > 0) is dense
and of full Hausdorff dimension N (D Dolgopyat,
V Kaloshin, and L Koralov, 2002).
dxt ¼ X0 ðxt Þ dt þ
See also: Bifurcation of Periodic Orbits (00027); Chaos
and Attractors (00093); Dissipative Dynamical Systems
of Infinite Dimension (00095); Dynamical Systems in
Mathematical Physics (00098); Ergodic Theory (00403);
Generic Properties of Dynamical Systems (00164);
Holomorphic Dynamics (00404); Homoclinic Phenomena
(00374); Hyperbolic Systems (00407); Lyapunov
Exponents, Strange Attractors (00100); Polygonal
Billiards (00452); Renormalization and the Feigenbaum
Phenomenon (00167); Stationary Solutions of PDEs: and
Heteroclinic/Homoclinic Connexions of Dynamical
Systems (00104); Synchronization of Chaos (00105);
Navier-Stokes Turbulence (00206); Partial Differential
Equations in Fluid Mechanics (00251); Quantum Chaos
(00332); Wavelets: Mathematical Theory (00153);
Mathematics of Image Processing (00367); Stochastic
Differential Equations (00369).
Other Directions
There are many other fractal dimensions important
for dynamics, like the Rényi spectrum for dimensions, correlation dimension, information dimension, 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,
CJ Bishop, PW Jones, C McMullen, BO Stratmann,
etc.), quasiconformal mappings and quasiconformal
groups (FW Gehring, J Väisäla, K Astala, CJ Bishop,
P Tukia, JW Anderson, P Bonfert-Taylor, EC Taylor,
etc.), graph directed Markov systems (RD Mauldin,
M Urbański, etc.), random walks on fractal graphs
(J Kigami, A Telcs, etc.), billiards (H Masur,
Y Cheung, P Bálint, S Tabachnikov, N Chernov,
D Szász, IP Tóth, etc.), quantum dynamics
(J-M Barbaroux, J-M Combes, H Schulz-Baldes,
I Guarneri, etc.), quantum gravity (M Aizenman,
A Aharony, ME Cates, TA Witten, GF Lawler,
B Duplantier, etc.), harmonic analysis (RS Strichartz,
ZM Balogh, JT Tyson, etc.), number theory
(L Barreira, M Pollicott, H Weiss, B Stratmann,
B Saussol, etc.), Markov processes (RM Blumenthal,
R Getoor, SJ Taylor, S Jaffard, C Tricot, Y Peres,
Y Xiao, etc.), and theoretical computer science
(B Ya Ryabko, L Staiger, JH Lutz, E Mayordomo,
etc.), and so on.
uniform dimension doubling property (R Kaufman,
1969). This means that for a.e. Brownian motion !
there holds dimH !(A) = 2 dimH 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 Lévy stable processes.
Fractal dimensions also appear in the study of
stochastic differential equations, like
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