C^m Eigenfunctions of Perron-Frobenius Operators and a New Approach to Numerical Computation of Hausdorff Dimension

0.81 MB·

by  Richard S. Falk

Cm EIGENFUNCTIONS OF PERRON-FROBENIUS OPERATORS
AND A NEW APPROACH TO NUMERICAL COMPUTATION
OF HAUSDORFF DIMENSION
6
RICHARDS.FALKANDROGERD.NUSSBAUM
1
0
2
n Abstract. WedevelopanewapproachtothecomputationoftheHausdorff
a dimensionoftheinvariantsetofaniteratedfunctionsystemorIFS.Intheone
J dimensional case, our methods require only C3 regularity of the maps in the
5 IFS.Thekeyidea,whichhasbeenknowninvaryingdegreesofgeneralityfor
2 manyyears,istoassociatetotheIFSaparametrizedfamilyofpositive,linear,
Perron-FrobeniusoperatorsLs. TheoperatorsLs cantypicallybestudiedin
] many different Banach spaces. Here, unlike most of the literature, we study
T Ls in a Banach space of real-valued, Ck functions, k ≥ 2; and we note that
N Ls is not compact, but has a strictly positive eigenfunction vs with positive
eigenvalue λs equal to the spectral radius of Ls. Under appropriate assump-
.
h tions on the IFS, the Hausdorff dimension of the invariant set of the IFS is
t the value s=s∗ for which λs =1. This eigenvalue problem is then approxi-
a
matedbyacollocationmethodusingcontinuouspiecewiselinearfunctions(in
m
onedimension)orbilinearfunctions(intwodimensions). Usingthetheoryof
[ positivelinearoperatorsandexplicitaprioriboundsonthederivativesofthe
strictlypositiveeigenfunctionvs,wegiverigorousupperandlowerboundsfor
1 theHausdorffdimensions∗,andtheseboundsconvergetos∗ asthemeshsize
v approacheszero.
7
3
7
6
0
1. Introduction
.
1
0
6 Our interest in this paper is in finding rigorous estimates for the Hausdorff
1 dimensionofinvariantsetsfor(possiblyinfinite)iteratedfunctionsystemsorIFS’s.
: The case of graph directed IFS’s (see [40] and [39]) is also of great interest and can
v
i bestudiedbyourmethods,butforsimplicityweshallrestrictattentionheretothe
X
IFS case.
r
a
Let D ⊂ Rn be a nonempty compact set, ρ a metric on D which gives the
topology on D, and θ : D → D, 1 ≤ j ≤ m, a contraction mapping, i.e., a
j
Lipschitz mapping (with respect to ρ) with Lipschitz constant Lip(θ ), satisfying
j
Lip(θ ) := c < 1. If m < ∞ and the above assumption holds, it is known that
j j
there exists a unique, compact, nonempty set C ⊂ D such that C = ∪m θ (C).
j=1 j
The set C is called the invariant set for the IFS {θ |1 ≤ j ≤ m}. If m = ∞ and
j
sup{c |1 ≤ j ≤ m} = c < 1, there is a naturally defined nonempty invariant set
j
Date:December1,2015.
2000 Mathematics Subject Classification. Primary11K55,37C30;Secondary: 65D05.
Keywordsandphrases. Hausdorffdimension,positivetransferoperators,continuedfractions.
TheworkofthesecondauthorwassupportedbyNSFgrantDMS-1201328.
1
2 RICHARDS.FALKANDROGERD.NUSSBAUM
C ⊂D such that C =∪∞ θ (C), but C need not be compact. It is useful to note
j=1 j
that the Lipschitz condition of the IFS can be weakened, and we address this in a
subsequent section (cf. (H6.1) in Section 6).
Although we shall eventually specialize, it may be helpful to describe initially
somefunctionanalyticresultsinthegeneralityofthepreviousparagraph. LetH be
abounded,open,mildlyregularsubsetofRn andletCk(H¯)denotetherealBanach
spaceofCk real-valuedmaps,allofwhosepartialderivativesoforderν ≤k extend
continuouslytoH¯. ForagivenpositiveintegerN,assumethatb :H¯ →(0,∞)are
j
strictly positive CN functions for 1≤j ≤m<∞ and θ :H¯ →H¯, 1≤j ≤m, are
j
CN maps and contractions. For s>0 and integers k, 0≤k ≤N, one can define a
bounded linear map L :Ck(H¯)→Ck(H¯) by the formula
s,k
m
(cid:88)
(1.1) (L f)(x)= [b (x)]sf(θ (x)).
s,k j j
j=1
Linear maps like L are sometimes called positive transfer operators or Perron-
s,k
Frobenius operators and arise in many contexts other than computation of Haus-
dorffdimension: see,forexample,[1]. Ifr(L )denotesthespectralradiusofL ,
s,k s,k
thenλ =r(L )ispositiveandindependentofk for0≤k ≤N;andλ isanalge-
s s,k s
braicallysimpleeigenvalueofL withacorrespondingunique,normalizedstrictly
s,k
positiveeigenfunctionv ∈CN(H¯). Furthermore,themaps(cid:55)→λ iscontinuous. If
s s
σ(L )⊂C denotes the spectrum of the complexification of L , σ(L ) depends
s,k s,k s,k
on k, but for 1≤k ≤N,
(1.2) sup{|z|:z ∈σ(L )\{λ }}<λ .
s,k s s
If k =0, the strict inequality in (1.2) may fail. A more precise version of the above
result in stated in Theorem 5.1 of this paper and Theorem 5.1 is a special case of
resultsin[46]. Themethodofproofinvolvesideasfromthetheoryofpositivelinear
operators,particularlygeneralizationsoftheKre˘ın-Rutmantheoremtononcompact
linearoperators;see[32],[3],[53],[44],and[37]. Wedonotusethethermodynamic
formalism (see [49]) and often our operators cannot be studied in Banach spaces of
analytic functions.
The linear operators which are relevant for the computation of Hausdorff di-
mension comprise a small subset of the transfer operators described in (1.1), but
the analysis problem which we shall consider here can be described in the gen-
erality of (1.1) and is of interest in this more general context. We want to find
rigorous methods to estimate r(L ) accurately and then use these methods to
s,k
estimate s , where, in our applications, s will be the unique number s ≥ 0 such
∗ ∗
that r(L )=1. Under further assumptions, we shall see that s equals dim (C),
s,k ∗ H
the Hausdorff dimension of the invariant set associated to the IFS. This observa-
tion about Hausdorff dimension has been made, in varying degrees of generality by
many authors. See, for example, [6], [7], [5], [9], [10], [13], [18], [20], [22], [21], [24],
[25], [26], [27], [39], [38], [47], [49], [50], [51], and [54].
In the applications in this paper, H will always be a bounded open subset of
Rn for n = 1 or 2. When n = 1, we shall assume that H is a finite union of
bounded open intervals, that θ : H¯ → H¯ is a CN contraction mapping, where
j
N ≥ 3, (or more generally satisfies (H6.1)) and θ(cid:48)(x) (cid:54)= 0 for all x ∈ H¯. In the
j
COMPUTATION OF HAUSDORFF DIMENSION 3
notation of (1.1), we define b (x) = |θ(cid:48)(x)|. When n = 2, we assume that H is
j j
a bounded, open mildly regular subset of R2 = C and that θ , 1 ≤ j ≤ m are
j
analytic or conjugate analytic contraction maps (or more generally satisfy (H6.1)),
defined on an open neighborhood of H¯ and satisfying θ (H) ⊂ H. We define
j
Dθ (z) = lim |[θ (z+h)−θ (z)]/h|, where h ∈ C in the limit, and we assume
j h→0 j j
that Dθ (z) (cid:54)= 0 for z ∈ H¯. In this case, L is defined by (1.1), with x replaced
j s,k
by z, and b (z)=|Dθ (z)|s.
j j
Given the existence of a strictly positive CN eigenfunction v for (1.1) when
s
H ⊂R,weshowinSection6forp=1andp=2,thatonecanobtainexplicitupper
and lower bounds for the quantity Dpv (x)/v (x) for x∈H¯, where Dp denotes the
s s
p-th derivative of v . Such bounds can also be obtained for p = 3 and p = 4, but
s
the arguments and calculations are more complicated. When H ⊂ R2, it is also
possible to obtain explicit upper and lower bounds for Dpv (x ,x ))/v (x ,x )
1 s 1 2 s 1 2
and Dpv (x ,x ))/v (x ,x ), where D = ∂/∂x and D = ∂/∂x . However, for
2 s 1 2 s 1 2 1 1 2 2
simplicity we restrict ourselves to the choice θ (z)=(z+β )−1, where β ∈C and
j j j
Re(β )>0. InthiscaseweobtaininSection7explicitupperandlowerboundsfor
j
Dpv (x ,x ))/v (x ,x )for1≤p≤4, 1≤k ≤2, andx >0. Inboththeoneand
k s 1 2 s 1 2 1
two dimensional cases, these estimates play a crucial role in allowing us to obtain
rigorous upper and lower bounds for the Hausdorff dimension.
ThebasicideaofournumericalschemeistocoverH¯ bynonoverlappingintervals
of length h if H ⊂ R or by nonoverlapping squares of side h if H ⊂ R2. We then
approximate the strictly positive, C2 eigenfunction v by a continuous piecewise
s
linear function (if H ⊂R) or a continuous piecewise bilinear function (if H ⊂R2).
Using the explicit bounds on the unmixed derivatives of v of order 2, we are then
s
able to associate to the operator L , square matrices A and B , which have
s,k s s
nonnegative entries and also have the property that r(A ) ≤ λ ≤ r(B ). A key
s s s
role here is played by an elementary fact which is not as well known as it should
be. If M is a nonnegative matrix and v is a strictly positive vector and Mv ≤λv,
(coordinate-wise), then r(M) ≤ λ. An analogous statement is true if Mv ≥ λv.
We emphasize that our approach is robust and allows us to study the case H ⊂R
when θ (·), 1≤j ≤m, is only C3.
j
If s denotes the unique value of s such that r(L )=λ =1, so that s is the
∗ s∗ s∗ ∗
Hausdorff dimension of the invariant set for the IFS under study, we proceed as
follows. Ifwecanfindanumbers suchthatr(B )≤1,then,sincethemaps(cid:55)→λ
1 s1 s
is decreasing, λ ≤r(B )≤1, and we can conclude that s ≤s . Analogously, if
s1 s1 ∗ 1
we can find a number s such that r(A )≥1, then λ ≥r(A )≥1, and we can
2 s2 s2 s2
conclude that s ≥s . By choosing the mesh size for our approximating piecewise
∗ 2
polynomials to be sufficiently small, we can make s −s small, providing a good
1 2
estimate for s . For a given s, r(A ) and r(B ) are easily found by variants of
∗ s s
the power method for eigenvalues, since (see Section 8) the largest eigenvalue has
multiplicityoneandistheonlyeigenvalueofitsmodulus. WhentheIFSisinfinite,
the procedure is somewhat more complicated, and we include the necessary theory
to deal with this case.
If the coefficients b (·) and the maps θ (·) in (1.1) are CN with N > 2, it
j j
is natural to approximate v (·) by piecewise polynomials of degree N −1 when
s
4 RICHARDS.FALKANDROGERD.NUSSBAUM
H ⊂ R and by corresponding higher order approximations when H ⊂ R2. The
corresponding matrices A and B may no longer have all nonnegative entries and
s s
theargumentsofthispaperare nolongerdirectlyapplicable. However, wehopeto
prove in a future paper that the inequality r(A ) ≤ λ ≤ r(B ) remains true and
s s s
leads to much improved upper and lower bounds for r(L ). Heuristic evidence for
s
this assertion is given in Table 3.2 of Section 3.2.
WeillustrateournewapproachbyfirstconsideringinSection3thecomputation
oftheHausdorffdimensionofinvariantsetsin[0,1]arisingfromclassicalcontinued
fraction expansions. In this much studied case, one defines θ = 1/(x+m), for
m
m a positive integer and x ∈ [0,1]; and for a subset B ⊂ N, one considers the
IFS {θ |m∈B} and seeks estimates on the Hausdorff dimension of the invariant
m
set C = C(B) for this IFS. This problem has previously been considered by many
authors. See [4], [6], [7], [18], [20], [22], [21], [25], [26], and [19]. In this case, (1.1)
becomes
(cid:88) (cid:16) 1 (cid:17)2s (cid:16) 1 (cid:17)
(L v)(x)= v , 0≤x≤1,
s,k x+m x+m
m∈B
and one seeks a value s ≥ 0 for which λ := r(L ) = 1. Table 3.1 in Section 3.2
s s,k
gives upper and lower bounds for the value s such that λ = 1 for various sets
s
B. Jenkinson and Pollicott [26] use a completely different method and obtain,
when |B| is small, high accuracy estimates for dim (C(B)), in which successive
H
approximations converge at a super-exponential rate. It is less clear (see [25]) how
welltheapproximationschemein[26]or[25]workswhen|B|ismoderatelylargeor
when different real analytic functions θˆ : [0,1] → [0,1] are used. Here, in the one
j
dimensionalcase,wepresentanalternativeapproachwithmuchwiderapplicability
that only requires the maps in the IFS to be C3. As an illustration, we consider in
Section 3.3 perturbations of the IFS for the middle thirds Cantor set for which the
corresponding contraction maps are C3, but not C4.
In Section 4, we consider the computation of the Hausdorff dimension of some
invariant sets arising for complex continued fractions. Suppose that B is a subset
of I = {m+ni|m ∈ N,n ∈ Z}, and for each b ∈ B, define θ (z) = (z +b)−1.
1 b
Note that θ maps G¯ = {z ∈ C||z−1/2| ≤ 1/2} into itself. We are interested in
b
the Hausdorff dimension of the invariant set C = C(B) for the IFS {θ |b ∈ B}.
b
This is a two dimensional problem and we allow the possibility that B is infinite.
In general (contrast work in [26] and [25]), it does not seem possible in this case
to replace L , k ≥ 2, by an operator Λ acting on a Banach space of analytic
s,k s
functions of one complex variable and satisfying r(Λ )=r(L ). Instead, we work
s s,k
inC2(G¯)andapplyourmethodstoobtainrigorousupperandlowerboundsforthe
Hausdorff dimension dim (C(B)) for several examples. The case B =I has been
H 1
of particular interest and is one motivation for this paper. In [16], Gardner and
Mauldin proved that d:=dim (C(I ))<2, in [38], Mauldin and Urbanski proved
H 1
that d ≤ 1.885, and in [48], Priyadarshi proved that d ≥ 1.78. In Section 4.2, we
prove that 1.85550≤d≤1.85589.
The square matrices A and B mentioned above and described in more detail
s s
in Section 3 have nonnegative entries and satisfy r(A ) ≤ λ ≤ r(B ). To apply
s s s
standard numerical methods, it is useful to know that all eigenvalues µ (cid:54)= r(A )
s
of A satisfy |µ| < r(A ) and that r(A ) has algebraic multiplicity one and that
s s s
COMPUTATION OF HAUSDORFF DIMENSION 5
corresponding results hold for r(B ). Such results are proved in Section 8 in the
s
one dimensional case when the mesh size, h, is sufficiently small, and a similar
argument can be used in the two dimensional case. Note that this result does not
followfromthestandardtheoryofnonnegativematrices, sinceA andB typically
s s
have zero columns and are not primitive. We also prove that r(A ) ≤ r(B ) ≤
s s
(1+C h2)r(A ), where the constant C can be explicitly estimated. In Section 9,
1 s 1
we prove that the map s(cid:55)→λ is log convex and strictly decreasing; and the same
s
result is proved for s (cid:55)→ r(M ), where M is a naturally defined matrix such that
s s
A ≤M ≤B .
s s s
Although many of the key results in the paper are described above, the paper
is long and an outline summarizing the sections may be helpful. In Section 2,
we recall the definition of Hausdorff dimension and present some mathematical
preliminaries. In Sections 3 and 4, we present the details of our approximation
scheme for Hausdorff dimension, explain the crucial role played by estimates on
derivatives of order ≤2 of v , and give the aforementioned estimates for Hausdorff
s
dimension. We emphasize that this is a feasibility study. We have limited the
accuracyofourapproximationstowhatiseasilyfoundusingthestandardprecision
of Matlab and have run only a limited number of examples, using mesh sizes that
allow the programs to run fairly quickly. In addition, we have not attempted to
exploit the special features of our problems, such as the fact that our matrices are
sparse. Thus, it is clear that one could write a more efficient code that would also
speed up the computations. However, the Matlab programs we have developed are
available on the web at www.math.rutgers.edu/~falk/hausdorff/codes.html,
and we hope other researchers will run other examples of interest to them.
The theory underlying the work in Sections 3 and 4 is deferred to Sections 5–9.
In Section 5 we describe some results concerning existence of Cm positive eigen-
functions for a class of positive (in the sense of order-preserving) linear operators.
We remark that Theorem 5.1 in Section 5 was only proved in [46] for finite IFS’s.
As a result, some care is needed in dealing with infinite IFS’s: see Theorem 5.2
and Corollary 5.3. In Section 6, we derive explicit bounds on the derivatives of
the eigenfunction v of L in the one-dimensional case and in Section 7, we derive
s s
explicit bounds on the derivatives of eigenfunctions of operators in which the map-
pings θ are given by M¨obius transformations which map a given bounded open
β
subsetH ofC:=R2 intoH. InSection8, weverifysomespectralpropertiesofthe
approximatingmatriceswhichjustifystandardnumericalalgorithmsforcomputing
their spectral radii. Finally, in Section 9, we show the log convexity of the spectral
radius r(L ), which we exploit in our numerical approximation scheme.
s
2. Preliminaries
We recall the definition of the Hausdorff dimension, dim (K), of a subset K ⊂
H
RN. To do so, we first define for a given s≥0 and each set K ⊂RN,
(cid:88)
Hs(K)=inf{ |U |s :{U } is a δ cover of K},
δ i i
i
where |U| denotes the diameter of U and a countable collection {U } of subsets of
i
RN is a δ-cover of K ⊂ RN if K ⊂ ∪ U and 0 < |U | < δ. We then define the
i i i
6 RICHARDS.FALKANDROGERD.NUSSBAUM
s-dimensional Hausdorff measure
Hs(K)= lim Hs(K).
δ
δ→0+
Finally, we define the Hausdorff dimension of K, dim (K), as
H
dim (K)=inf{s:Hs(K)=0}.
H
We now state the main result connecting Hausdorff dimension to the spectral
radius of the map defined by (1.1). To do so, we first define the concept of an
infinitesimalsimilitude. Let(S,d)beacompact,perfectmetricspace. Ifθ :S →S,
then θ is an infinitesimal similitude at t ∈ S if for any sequences (s ) and (t )
k k k k
with s (cid:54)=t for k ≥1 and s →t, t →t, the limit
k k k k
d(θ(s ),θ(t )
lim k k =:(Dθ)(t)
k→∞ d(sk,tk)
existsandisindependentoftheparticularsequences(s ) and(t ) . Furthermore,
k k k k
θ is an infinitesimal similitude on S if θ is an infinitesimal similitude at t for all
t∈S.
This concept generalizes the concept of affine linear similitudes, which are affine
linear contraction maps θ satisfying for all x,y ∈Rn
d(θ(x),θ(y))=cd(x,y), c<1.
Inparticular,theexamplesdiscussedinthispaper,suchasmapsoftheformθ(x)=
1/(x+m),withmapositiveinteger,areinfinitesimalsimilitudes. Moregenerally,if
S isacompactsubsetofR1 andθ :S →S extendstoaC1 mapdefinedonanopen
neighborhood of S in R1, then θ is an infinitesimal similitude. If S is a compact
subset of R2 :=C and θ :S →S extends to an analytic or conjugate analytic map
defined on an open neighborhood of S in C, θ is an infinitesimal similitude.
Theorem2.1. (Theorem1.2of[47].) Letθ :S →S for1≤i≤N beinfinitesimal
i
similitudes and assume that the map t (cid:55)→ (Dθ )(t) is a strictly positive H¨older
i
continuousfunctiononS. Assumethatθ isaLipschitzmapwithLipschitzconstant
i
c ≤c<1 and let C denote the unique, compact, nonempty invariant set such that
i
C =∪N θ (C).
i=1 i
Further, assume that θ satisfy
i
θ (C)∩θ (C)=∅, for 1≤i,j ≤N. i(cid:54)=j
i j
andareone-to-oneonC. ThentheHausdorffdimensionofC isgivenbytheunique
σ such that r(L )=1.
0 σ0
3. Examples in one dimension
3.1. Continued fraction Cantor sets. Wefirstconsidertheproblemofcomput-
ing the Hausdorff dimension of some Cantor sets arising from continued fraction
COMPUTATION OF HAUSDORFF DIMENSION 7
expansions. More precisely, given any number 0 < x < 1, we can consider its
continued fraction expansion
1
x=[a ,a ,a ,...]= ,
1 2 3
1
a +
1
1
a +
2 a +···
3
where a ,a ,a ,...∈N. We then consider the Cantor set E , of all points
1 2 3 [m1,...,mp]
in [0,1] where we restrict the coefficients a to the values m ,...,m . A number of
i 1 p
papers(e.g.,[6],[7],[18],[20],[22],[26])haveconsideredthisprobleminthecaseof
the set E , consisting of all points in [0,1] for which each a has the value 1 or 2.
1,2 i
In [26], a method is presented that computes this dimension to 25 decimal places.
Computations are also presented in that paper and in [25] for other choices of the
values m ,...,m . In [4], the Hausdorff dimension of the Cantor set E is
1 p 2,4,6,8,10
computed to three decimal places (0.517).
Corresponding to the choices of m , we associate contraction maps θ (x) =
i m
1/(x+m). AkeyfactisthattheCantorsetsweconsidercanbegeneratedaslimit
points of sequences of these contraction maps. For example, the set E can be
1.2
generated using the maps θ (x) = 1/(x+1) and θ (x) = 1/(x+2) as the set of
1 2
limit points of sequences θ ...θ (0), for m ,m ,...∈{1,2}.
m1 mn 1 2
For v ∈C[0,1], we define
p
(cid:88)(cid:12) (cid:12)s
(3.1) (L v)(x)= (cid:12)θ(cid:48) (x)(cid:12) v(θ (x)).
s (cid:12) mj (cid:12) mj
j=1
Our computations are based on the following result, which we shall prove in subse-
quent sections.
Theorem 3.1. For all s>0, L has a unique strictly positive eigenvector v with
s s
L v = λ v , where λ > 0 and λ = r(L ), the spectral radius of L . Further-
s s s s s s s s
more, the map s (cid:55)→ λ is strictly decreasing and continuous, and for all p > 0,
s
(−1)pD(p)v (x)>0 for all x∈[0,1] and
s
(3.2) |D(p)v (x)|≤(2s)(2s+1)···(2s+p−1)(γ−p)v (x),
s s
where γ = min m . Finally, the Hausdorff dimension of the Cantor set generated
j j
from the maps θ , ..., θ is the unique value of s with λ =1.
m1 mp s
Note that it follows easily from (3.2) when p=1 and x ,x ∈[0,1] , that
1 2
(3.3) v (x )≤v (x )exp(2s|x −x |/γ).
s 2 s 1 2 1
To see this, write
v (x ) (cid:90) x2 d (cid:90) x2 v(cid:48)(x)
log s 2 =logv (x )−logv (x )= logv (x)dx= s dx,
v (x ) s 2 s 1 dx s v (x)
s 1 x1 x1 s
apply the bound in (3.2), and exponentiate the result.
To obtain approximations of the dimension of the Cantor sets described in this
section, we first approximate a function f ∈ C2[0,1] by a continuous, piecewise
linear function defined on a mesh of interval size h on [0,1]. More specifically, we
8 RICHARDS.FALKANDROGERD.NUSSBAUM
approximate f(x), x ≤ x ≤ x by its piecewise linear interpolant fI(x) given
k k+1
by
x −x x−x
fI(x)= k+1 f(x )+ kf(x ), x ≤x≤x ,
h k h k+1 k k+1
where the mesh points x satisfy 0 = x < x ,··· < x = 1, with x −x =
k 0 1 N k+1 k
h = 1/N. The goal is to reduce the infinite dimensional eigenvalue problem to a
finite dimensional one. Standard results for the error in linear interpolation on an
interval [a,b] assert that
1
fI(x)−f(x)= (b−x)(x−a)f(cid:48)(cid:48)(ξ)
2
for some ξ ∈[a,b]. If x ≤θ (x)≤x , we get
rj mj rj+1
[x −θ (x)] [θ (x)−x ]
vI(θ (x))= rj+1 mj v (x )+ mj rj v (x ).
s mj h s rj h s rj+1
We can also use the properties in Theorem 3.1 to bound the interpolation error.
Letting f(x)=v (x), we obtain from Theorem 3.1 that
s
0<v(cid:48)(cid:48)(θ (x))≤2s(2s+1)γ−2v (θ (x)).
s mj s mj
Using the interpolation error estimate and (3.3), we get for x ≤θ (x)≤x ,
rj mj rj+1
0<vI(θ (x))−v (θ (x))
s mj s mj
≤[x −θ (x)][θ (x)−x ]s(2s+1)γ−2 max v (ξ).
rj+1 mj mj rj [xrj,xrj+1]s
≤[x −θ (x)][θ (x)−x ]s(2s+1)γ−2exp(2sh/γ)vI(θ (x)),
rj+1 mj mj rj s mj
since the point at which the maximum occurs is within h of either of the two
endpoints of the subinterval.
Using this estimate, we have precise upper and lower bounds on the error in the
interval [x ,x ] that only depend on the function values of v at x and x .
rj rj+1 s rj rj+1
Letting
err (x)=[x −θ (x)][θ (x)−x ]s(2s+1)γ−2exp(2sh/γ),
j rj+1 mj mj rj
we have for each mesh point x , with x ≤θ (x )≤x ,
k rj mj k rj+1
[1−err (x )]vI(θ (x ))≤v (θ (x ))≤vI(θ (x )).
j k s mj k s mj k s mj k
Since for each mesh point x , r(L )v (x ) = (L v )(x ), we can use (3.1) and
k s s k s s k
the above result to to see that
p
(cid:88)(cid:12) (cid:12)s
r(L )v (x )=L v (x )= (cid:12)θ(cid:48) (x )(cid:12) v (θ (x ))
s s k s s k (cid:12) mj k (cid:12) s mj k
j=1
p
(cid:88)(cid:12) (cid:12)s
≤ (cid:12)θ(cid:48) (x )(cid:12) vI(θ (x ))
(cid:12) mj k (cid:12) s mj k
j=1
and
p
(cid:88)(cid:12) (cid:12)s
r(L )v (x )≥ (cid:12)θ(cid:48) (x )(cid:12) [1−err (x )]vI(θ (x )).
s s k (cid:12) mj k (cid:12) j k s mj k
j=1
COMPUTATION OF HAUSDORFF DIMENSION 9
Let w be a vector with (w ) = v (x ), k = 0,...N. Define (N +1)×(N +1)
s s k s k
matrices B and A by
s s
p
(cid:88)(cid:12) (cid:12)s
(B w ) = (cid:12)θ(cid:48) (x )(cid:12) wI(θ (x )),
s s k (cid:12) mj k (cid:12) s mj k
j=1
p
(cid:88)(cid:12) (cid:12)s
(A w ) = (cid:12)θ(cid:48) (x )(cid:12) [1−err (x )]wI(θ (x )),
s s k (cid:12) mj k (cid:12) j k s mj k
j=1
where, if x ≤θ (x)≤x , we define
rj mj rj+1
[x −θ (x)] [θ (x)−x ]
wI(θ (x))= rj+1 mj (w ) + mj rj (w ) .
s mj h s rj h s rj+1
Note that all of the entries of B will be nonnegative and since err (x) = O(h2),
s j
this is true for A as well, provided h is sufficiently small.
s
Since v (x ) > 0 for k = 0,...,N, we can apply the following result about
s k
nonnegative matrices to see that
r(A )≤r(L )≤r(B ).
s s s
Lemma 3.2. Let M be an (N+1)×(N+1) matrix with non-negative entries and
w an N +1 vector with strictly positive components.
If (Mw) ≥λw , k =0,...N, then r(M)≥λ,
k k
If (Mw) ≤λw , k =0,...N, then r(M)≤λ.
k k
Since this result is crucial to our approximation scheme, we supply the proof
below to keep our presentation self-contained. Note, however, that Lemma 3.2 is
actually a special case of much more general results concerning order-preserving,
homogeneousconemappings: seeLemma2.2in[34]andTheorem2.2in[36]. Ifwe
let D denote the positive diagonal (N +1)×(N +1) matrix with diagonal entries
w , 1 ≤ j ≤ N +1, r(M) = r(D−1MD); and Lemma 3.2 can also be obtained by
j
applying Theorem 1.1 on page 24 of [41] to D−1MD.
Proof. If (Mw) ≥ λw , k = 0,...N, it easily follows that (Mnw) ≥ λnw and
k k k k
so (cid:107)Mnw(cid:107) ≥λn(cid:107)w(cid:107) . Let e be vector with all e =1. Then
∞ ∞ i
(cid:107)Mn(cid:107) =(cid:107)Mne(cid:107) ≥(cid:107)Mnw(cid:107) /(cid:107)w(cid:107) ≥λn.
∞ ∞ ∞ ∞
Hence,
r(M)= lim (cid:107)Mn(cid:107)1/n ≥λ.
∞
n→∞
If (Mw) ≤ λw , k = 0,...N, it easily follows that (Mnw) ≤ λnw . Let k be
k k k k
chosen so that (cid:107)Mn(cid:107) =(cid:80) (Mn) . Since [r(M)]n =r(Mn)≤(cid:107)Mn(cid:107) ,
∞ j k,j ∞
(cid:88) (cid:88)
minw [r(M)]n ≤minw (Mn) ≤ (Mn) w =(Mnw) ≤λnw .
j j k,j k,j j k k
j j
j j
So,
minw ≤[λ/r(M)]nw .
j k
j
If r(M) > λ, then letting n → ∞, we get that min w ≤ 0, which contradicts the
j j
fact that all w >0. Hence, r(M)≤λ. (cid:3)
j
10 RICHARDS.FALKANDROGERD.NUSSBAUM
As described in Section 1, if s denotes the unique value of s such that r(L )=
∗ s∗
λ =1, then s is the Hausdorff dimension of the set E . If we can find a
s∗ ∗ [m1,...,mp]
number s such that r(B ) ≤ 1, then r(L ) ≤ r(B ) ≤ 1, and we can conclude
1 s1 s1 s1
that s ≤ s . Analogously, if we can find a number s such that r(A ) ≥ 1, then
∗ 1 2 s2
r(L ) ≥ r(A ) ≥ 1, and we can conclude that s ≥ s . By choosing the mesh
s2 s2 ∗ 2
sufficiently fine, we can make s −s small, providing a good estimate for s .
1 2 ∗
We can also reduce the number of computations by first iterating the maps θ
mi
to produce a smaller initial domain that we need to approximate. For example,
if we seek the Hausdorff dimension of the set E , since θ ([0,1]) = [1/2,1] and
1,2 1
θ ([0,1]) = [1/3,1/2], the maps θ and θ map [1/3,1] (cid:55)→ [1/3,1], so we can re-
2 1 2
strict the problem to this subinterval. Further iterating, we see that θ ([1/3,1])=
1
[1/2,3/4] and θ ([1/3,1])=[1/3,3/7]. Hence the maps θ and θ map [1/3,3/7]∪
2 1 2
[1/2,3/4] to itself and we can further restrict the problem to this domain.
3.2. Continued fraction Cantor sets – numerical results. Inthissection, we
report in Table 3.1 the results of the application of the algorithm described above
to the computation of the Hausdorff dimension of a sample of continued fraction
Cantor sets. Where the true value was known to sufficient accuracy, it is not hard
to check that the rate of convergence as h is refined is O(h2). Although the theory
developed above does not apply to higher order piecewise polynomial approxima-
tion, since one cannot guarantee that the approximate matrices have nonnegative
entries, we also report in Table 3.2 and Table 3.3 the results of higher order piece-
wise polynomial approximation to demonstrate the promise of this approach. In
thiscase,weonlyprovidetheresultsforB ,whichdoesnotcontainanycorrections
s
for the interpolation error. In a future paper we hope to prove that rigorous upper
and lower bounds for the Hausdorff dimension can also be obtained when higher
order piecewise polynomial approximations are used.
The errors are computed based on the results reported in [26]. For the last
five entries, we do not have independent results for the true solution correct to a
sufficient number of decimal places to compute the error.
In the computations shown using higher order piecewise polynomials, since the
number of unknowns for a continuous, piecewise polynomial of degree k on N uni-
formly spaced subintervals of width h is given by kN+1, to get a fair comparison,
wehaveadjustedthemeshsizessothateachcomputationinvolvesthesamenumber
of unknowns. For this problem, the eigenfunction v is smooth and the computa-
s
tions show a dramatic increase in the accuracy of the approximation as the degree
of the approximating piecewise polynomial is increased.
3.3. An example with less regularity. For 0≤λ≤1, we consider the maps
1 1 2+λ
(3.4) θ (x)= (x+λx7/2), θ (x)= (x+λx7/2)+ ,
1 3+2λ 2 3+2λ 3+2λ
which map the unit interval to itself. Both these maps ∈C3([0,1], but ∈/ C4([0,1].
We note that because of the lack of regularity, the methods of [26] and [25] cannot
be applied. When λ=0, these maps become
x x 2
θ (x)= , θ (x)= + ,
1 3 2 3 3