This article was downloaded by: [Macquarie University]On: 28 October 2014, At: 17:53Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T3JH, UK

Dynamical Systems: AnInternational JournalPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/cdss20

An adaptive subdivisiontechnique for theapproximation of attractorsand invariant measures: proofof convergenceOliver JungePublished online: 16 Jul 2010.

To cite this article: Oliver Junge (2001) An adaptive subdivision techniquefor the approximation of attractors and invariant measures: proof ofconvergence, Dynamical Systems: An International Journal, 16:3, 213-222, DOI:10.1080/14689360109696233

To link to this article: http://dx.doi.org/10.1080/14689360109696233

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all theinformation (the “Content”) contained in the publications on our platform.However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness,or suitability for any purpose of the Content. Any opinions and viewsexpressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of theContent should not be relied upon and should be independently verifiedwith primary sources of information. Taylor and Francis shall not be liablefor any losses, actions, claims, proceedings, demands, costs, expenses,damages, and other liabilities whatsoever or howsoever caused arisingdirectly or indirectly in connection with, in relation to or arising out of theuse of the Content.

This article may be used for research, teaching, and private studypurposes. Any substantial or systematic reproduction, redistribution,reselling, loan, sub-licensing, systematic supply, or distribution in any formto anyone is expressly forbidden. Terms & Conditions of access and use canbe found at http://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

Mac

quar

ie U

nive

rsity

] at

17:

53 2

8 O

ctob

er 2

014

Dynamical Systems, Vol. 16, No. 3, 2001, 213± 222

An adaptive subdivision technique for the approximation ofattractors and invariant measures: proof of convergence

OLIVER JUNGE

Department of Mathematics and Computer Science, University of

Paderborn, 33098 Paderborn, Germany

e-mail: junge@upb.de

Abstract. We prove convergence of a recently introduced adaptive multilevel algor-ithm for the e� cient computation of invariant measures and attractors of dynamical

systems. The proof works in the context of (su� ciently regular) stochastic processes

and essentially shows that the discretization of phase space leads to a small random

perturbation of the original process. A generalized version of a lemma of

Khasminskii then gives the desired result.

Received 22 March 2001

1. General introduction

The statistics of the long time behaviour of a dynamical system are captured by

certain invariant measures’ . These measures weight a part of the state space accord-

ing to the probability by which a typical trajectory of the system is being observed in

this part. In this sense these `natural invariant’ (or SRB-) measures are objects of

fundamental interest in the study of the long time behaviour of dynamical systems.

In easy cases, an approximation to one of these natural invariant measures can be

obtained by computing a histogram of a (very) long trajectory of the system: one

counts the number of times that the trajectory enters a certain part of phase space.

Unfortunately this method lacks results on the connection between the length of the

orbit and the accuracy of the approximated invariant measure.

An alternative approach, commonly referred to as `Ulam’s method’ is to partition

(part of) the phase space into a ® nite number of sets and to employ the dynamical

system under consideration to compute `transition probabilities’ between the ele-

ments of the partition. This yields a Markov model of the original system in the

form of a stochastic matrix. An approximate invariant measure is then given by an

invariant vector of this matrix.

In this paper we give a proof of convergence for an adaptive version of Ulam’s

method. The idea of the algorithm (Dellnitz and Junge, 1998) is to use the informa-

tion provided by the invariant measure to selectively re® ne part of the partition in

one step of the algorithm. After a certain number of steps an adaptive partition has

been constructed and typically the estimate of the invariant measure is signi® cantly

Dynamical Systems ISSN 1468± 9367 print/ISSN 1468± 9375 online # 2001 Taylor & Francis Ltdhttp:/ /www.tandf.co.uk/journals

DOI: 10.1080/14689360110060708

Dow

nloa

ded

by [

Mac

quar

ie U

nive

rsity

] at

17:

53 2

8 O

ctob

er 2

014

better than the estimate based on a uniform partition of the same cardinalityÐ see

the example in section 4.

The algorithm described here has been implemented and is available within the

software package GAIO.1

2. Background

Ulam (1960) proposed the following method for an approximate computation of an

invariant measure of a map f : ‰0; 1Š ! ‰0; 1Š. Divide the unit interval into k intervals

I1; . . . ; Ik of equal length and consider the matrix

…pij† ˆm…Ij \ f 1…Ii††

m…Ij†;

i; j ˆ 1; . . . ; k, of `transition probabilities’ between the intervals. This matrix has a

largest eigenvalue 1 and Ulam conjectured that for certain maps the step function hk

de® ned by the eigenvector to this eigenvalue provides an approximation to the

density of an invariant measure of f . Li (1976) showed that indeed, if f is a piecewise

expanding C2-map possessing a unique invariant density h, then hk converges in L1

to h as k ! 1. Li exploited the fact that the transition matrix arises from a dis-

cretization of the `Frobenius± Perron operator’ , which for this particular class of map

can be shown to be su� ciently regular (see e.g. Keller (1982)). This result has been

generalized in various directions, see for example Boyarsky and Lou (1991), Chiu etal. (1992), Boyarsky et al. (1994), Miller (1994), Froyland (1995, 1996), Hunt (1996),

Murray (1997), Blank and Keller (1998), and Dellnitz and Junge (1999).

In order to improve the e� ciency of the method, Ding et al. (1993) suggested using

ansatzfunctions of higher order instead of piecewise constant functions. However, to

get a higher rate of convergence this approach essentially requires the unknown

invariant density to exhibit the same smoothness properties as the ansatzfunctions.

In Dellnitz and Junge (1998) another property of Ulam’s original method was modi-® ed. Instead of requiring the partition sets to be of equal size it was proposed to

locally adapt the size of the partition sets in a suitable way to the invariant measure.

To be more precise the therein advocated `adaptive subdivision algorithm’ used the

information provided by an approximate invariant measure in each step to selec-

tively re® ne part of the partition for the next step. The numerical experiments in thispaper suggested that in certain cases this algorithm delivers a higher rate of conver-

gence than Ulam’s `standard’ method.

The aim of the present paper is to give a convergence result for this new algorithm

in the context of su� ciently regular stochastic transition functions. The advantage of

considering transition functions is that as in Dellnitz and Junge (1999) the conver-gence result can directly be combined with results on the stochastic stability of, for

example, SRB-measures of uniformly hyperbolic systems (Kifer 1986) in order to

obtain convergence results for these systems. Now a crucial property of the approach

in Dellnitz and Junge (1999) (and the conventional `Ulam type’ methods) is that the

underlying partition of the phase space is uniformly re® ned in order to obtain

214 O. Junge

1http://www.upb.de/math/Ä agdellnitz/gaio

Dow

nloa

ded

by [

Mac

quar

ie U

nive

rsity

] at

17:

53 2

8 O

ctob

er 2

014

convergence. Since this is the very property missing in the adaptive approach the

proof in Dellnitz and Junge (1999) does not directly carry over.

The proof we give here (which is also contained in the thesis of Junge (1999))

centres around a generalization of a lemma due to Khasminskii (1963): consider a

small random perturbation’ of the original deterministic system f given by a familyof stochastic transition functions p". For every " let ·" be an invariant measure of p".

If a sequence …·"k†k with "k ! 0 as k ! 1 converges (weakly), then the limit is an

invariant measure for f . We are going to generalize the notion of a small random

perturbation to the context of stochastic transition functions and prove the analogue

to Khasminskii’ s result. Now in order to prove convergence of the adaptive subdivi-

sion algorithm we show that the discretization of the underlying stochastic processcan be interpreted as a small random perturbation in the generalized sense. Then the

application of (the generalized version of) Khasminskii’ s lemma leads to the desired

result.

The paper is organized as follows: after some introductory remarks in §3 on the

discretization of the Frobenius± Perron operator in our context we describe theadaptive subdivision algorithm in §4. In order to make the paper self-contained

and to give motivation for considering this particular algorithm we recall numerical

results obtained in Dellnitz and Junge (1998) in §4. Finally in §5 we state and prove

the convergence theorem.

3. Discretization of the Frobenius± Perron operator

We consider an evolution on the compact state space X » n de® ned by a transition

function p : X £ A ! ‰0; 1Š (A the Borel-¼-Algebra on X ): for every x 2 X the func-

tion p…x; ¢† is a probability measure on X and we assume that p is `strong Feller’ , i.e.

that the function x 7!„

g…y† p…x; dy† is continuous for every bounded, measurablefunction g : X ! . We suppose furthermore that X is forward invariant in the sense

that p…x; X† ˆ 1 for all x. The motivation to consider transition functions (instead of

some deterministic system itself) stems from the fact that there are well established

results on the stochastic stability of SRB-measures (see e.g. Kifer (1986)). In these

theorems the stochastically perturbed deterministic system is modelled by a (su� -ciently regular) stochastic transition function.

An invariant measure’ of a transition function p is a probability measure · on X

such that ·…A† ˆ„

p…x; A† ·…dx† for all measurable A. It is immediate that if we

de® ne the `Frobenius± Perron’ operator P : M ! M on the space M of probability

measures on X by

P·…A† ˆ…

p…x; A† ·…dx†; A 2 A;

then a measure is invariant iŒit is a ® xed point of P.

Now the basic idea underlying the approximate computation of invariant meas-

ures is to discretize the ® xed point problem’ P· ˆ · and to look for ® xed points of

the discretized operator (i.e. eigenvectors to the eigenvalue 1 of a corresponding

® nite dimensional matrix). To this end we introduce spaces of `discrete probability

measures’ as follows. Let B ˆ fB1; . . . ; Bbg be a ® nite partition (of a part of X), i.e. a® nite collection of compact subsets of X such that m…Bi \ Bj† ˆ 0 for i 6ˆ j, where m

denotes Lebesgue measure on X . The space MB of discrete probability measures on

B is the set of probability measures having a density (with respect to m) that is

215Approximation of attractors and invariant measures

Dow

nloa

ded

by [

Mac

quar

ie U

nive

rsity

] at

17:

53 2

8 O

ctob

er 2

014

piecewise constant on the elements of the partition B. Obviously a discrete prob-

ability measure · is determined by a non-negative vector u 2 b with kuk1 ˆ 1,

where ui ˆ ·…Bi†, i ˆ 1; . . . ; b. Using the projection QB : M ! MB

QB·…A† ˆX

B2B

m…A \ B†m…B† ·…B†;

we de® ne the `discretized Frobenius± Perron operator’ PB : MB ! MB, PB ˆ QBP.

4. The adaptive subdivision algorithm

The algorithm described below computes a sequence of pairs …B0; ·0†; …B1; ·1†; . . . ;where each Bk is a partition (of a subset of X) and each ·k is a discrete probability

measure invariant under PBk. Given …B0; ·0†, one inductively obtains …Bk; ·k† from

…Bk 1; ·k 1† for k ˆ 1; 2; . . . in two steps:

(i) Subdivision: choose ¯k 1 2 ‰0; 1Š, set B‡k 1 ˆ fB 2 Bk 1 : ·k 1…B† ¶ ¯k 1g and

construct a new collection BB‡k by re® ning the elements of B‡

k 1 such that

diam…BB‡k † µ ³ diam…B‡

k 1† for some 0 < ³ < 1.

(ii) Selection: Set BBk ˆ BB‡k [ …Bk 1nB‡

k 1†. Compute a ® xed point ·k of the dis-

cretized Frobenius± Perron operator PBBk. Set Bk ˆ fB 2 BBk : ·k…B† > 0g.

We terminate the process if minB2Bkdiam…B† < ­ for some k, where ­ > 0 is some

prescribed minimal diameter. For details concerning the numerical realization of the

algorithm we refer to Dellnitz and Junge (1998).

216 O. Junge

Figure 1. L1-error between hlog and the approximate invariant densities versus the

cardinality of the underlying partitions.

Dow

nloa

ded

by [

Mac

quar

ie U

nive

rsity

] at

17:

53 2

8 O

ctob

er 2

014

4.1. Numerical example

We recall a numerical result from Dellnitz and Junge (1998). Consider the logistic

map f …x† ˆ 4x…1 x† with invariant density hlog…x† ˆ …º������������������x…1 x†

p† 1 (see e.g.

Lasota and Mackey (1994)). We approximate hlog by Ulam’s standard method

using equipartitions and by applying the adaptive subdivision algorithm, choosing

¯k ˆ 1=bk, where bk is the number of boxes in Bk. The transition probabilities have

been computed by the method presented in Guder et al. (1997); for the computation

of the invariant probability vector we used an Arnoldi method (see Lehoucq et al.

(1998)) as implemented in the `eigs’ command in Matlab.

Figure 1 shows the L1-error between hlog and the approximate densities versus the

cardinality of the partitions. Note that since hlog is known the error between theapproximate densities and the true density can be computed exactly.

5. Proof of convergenceIn this section we are going to establish the following convergence result for the

adaptive subdivision algorithm.

Theorem 1. Let the transition function p…x; ¢† : A ! ‰0; 1Š be absolutely continuous

for all x with transition density h…x; ¢† : X ! and let h be uniformly bounded, i.e.

h…x; y† µ K for all x; y 2 X. If p has a unique invariant measure · and if ¯k ! 0 ask ! 1 then the sequence ·k of discrete probability measures generated by the algor-

ithm converges weakly to the invariant measure of p.

Note in particular that we require the invariant measure of p to be unique, su� -

cient conditions for this can be found in, for example Hunt (1996). As stated in the

introduction the main idea of the proof of Theorem 1 is to show that the discretized

Frobenius± Perron operator emerges as the transition operator of a transition func-tion which (under the hypotheses of the theorem) can be shown to be a small random

perturbation of the original transition function p. A generalized version of a lemma

of Khasminskii then immediately gives the desired result.

So let us begin with the generalized notion of a small random perturbation. For

ease of notation we are going to write

·…g† ˆ…

g d· and p…x; g† ˆ…

g…y†p…x; dy†

for measures · and measurable, bounded functions g : X ! .

De® nition 1. Let p be a transition function and pµ, µ > 0, be a family of transitionfunctions. If there is a family of probability measures ·µ, such that

…jp…x; g† pµ…x; g†j·µ…dx† ! 0

as µ ! 0 for continuous functions g : X ! , then the family pµ is called a smallrandom perturbation of p (with respect to the family ·µ).

The lemma of Khasminskii (1963) can immediately be generalized to this context.

Lemma 1. Consider a small random perturbation pµ of a transition function p with

respect to a family ·µ of probability measures. Suppose that p is strong Feller. If ·µ is

217Approximation of attractors and invariant measures

Dow

nloa

ded

by [

Mac

quar

ie U

nive

rsity

] at

17:

53 2

8 O

ctob

er 2

014

an invariant measure of pµ for each µ and ·µi! · weakly for some subsequence µi ! 0

and i ! 1, then · is an invariant measure for p.

Proof. Let g : X ! be a continuous function. Then

j·…g† …P·†…g†j µ j·…g† ·µi…g†j ‡ j…Pµi

·µi†…g† …P·µi

†…g†j

‡j…P·µi†…g† …P·†…g†j:

The ® rst term on the right-hand side vanishes for i ! 1 since we supposed ·µito

converge weakly to ·. Let us consider the third term. We have

…P·†…g† ˆ„

p…x; g† ·…dx†. Since p was supposed to be strong Feller, the map

x 7! p…x; g† is continuous. Using the weak convergence of ·µiagain, it follows that

the third term vanishes also as i ! 1. Finally we use the fact that pµ is a small

random perturbation of p in order to see that also the second term goes to zero.

Thus we get ·…g† ˆ …P·†…g†, and this relation being true for any continuous

function g implies that · is an invariant measure for p. &

The next step in the proof of Theorem 1 is to relate the discretized Frobenius±

Perron operator to some transition function which in turn will then be shown to be a

small random perturbation of p. In order to ® nd a suitable transition function wehave a closer look at the discretized Frobenius± Perron operator:

…PB·†…A† ˆ …QBP·†…A† ˆ… X

B2B

m…A \ B†m…B† p…x; B† ·…dx†: …1†

Thus PB is the transition operator of the transition function

pB…x; A† ˆX

B2B

m…A \ B†m…B†

p…x; B† ˆ…

A…y†X

B2B

B…y†m…B†

p…x; B† m…dy†

which will be the object of main interest further on.

Our goal now is to show that (under the hypotheses of Theorem 1) the family pBk

is a small random perturbation of p with respect to the family ·k (where Bk; ·k arethe partitions and measures generated by the adaptive subdivision algorithm).

According to De® nition 1 we have to show that, roughly speaking, asymptotically

the discrete transition distribution pBk…x; ¢† equals p…x; ¢† `for ¸-almost all x’ , where ¸

is an accumulation point of the sequence ·k. It is crucial to note that we do not have

to show asymptotic equality of pBk…x; ¢† and p…x; ¢† for all x, not even for Lebesgue

almost all x. Since at the end the accumulation point ¸ turns out to be an invariant

measure of p, we are in some sense using the information provided by the invariant

measure ¸ beforehand.

So we have to show that…

jp…x; g† pBk…x; g†j ·k…dx† ! 0 …2†

for k ! 1 and continuous functions g. Note that we can write…

g…y† pBk…x; dy† ˆ

……QBk

g†…y† p…x; dy†;

where we de® ne the projection QBkfrom L1 into the space of simple functions on the

partition Bk by

218 O. Junge

Dow

nloa

ded

by [

Mac

quar

ie U

nive

rsity

] at

17:

53 2

8 O

ctob

er 2

014

QBkg ˆ

X

B2Bk

B

m…B†

…

B

g dm.

Using this rewrite we estimate

jp…x; g† pBk…x; g†j µ

…jg…y† …QBk

g†…y†j p…x; dy† …3†

ˆ p…x; jg QBkgj†:

So abbreviating dk ˆ jg QBkgj, what we ® nally have to show is that

…p…x; dk† ·k…dx† ! 0 …4†

for k ! 1 and every continuous function g.

To this end we split up the partitions Bk into a `transient’ part T k and a support-ing’ part Sk. In T k we collect all partition sets that never get subdivided again or

have been removed in a previous selection step of the algorithm, that is

T k ˆ fB 2 Bk : ·`…B† < ¯` for ` ¶ kg [[k

jˆ1

BBjnBj:

Accordingly let Sk ˆ BknT k. We denote Tk ˆS

B2T kB; T ˆ

S1kˆ0 Tk; Sk ˆ

SB2Sk

B

and S ˆ XnT : We are now going to show (4) by splitting the integral into two parts:

p…x; dk† ˆ p…x; Tkdk† ‡ p…x; Sk

dk†.

5.1. The transient part T

Let ¸ be an accumulation point of the sequence …·k†k. In the following we treat a

corresponding convergent subsequence which, abusing notation, we shall also denoteby …·k†k. We treat the transient part by showing that the transition probability into

T is zero for ¸-almost all points.

Proposition 1. If ¯` ! 0 as ` ! 1 then p…x; T† ˆ 0 for ¸-almost all x 2 X.

Proof. It su� ces to show the assertion for any B 2 T k, k ˆ 0; 1; . . ., instead of T .

Since ·` is a ® xed point of PB`for all ` ˆ 0; 1; . . ., we have (using equation (1))

·`…B† ˆ …PB`·`†…B† ˆ

…p…x; B† ·`…dx†:

Since p is strong Feller the function x 7! p…x; B† is continuous. Thus

·`…B† !…

p…x; B† ¸…dx† as ` ! 1;

taking a suitable subsequence if necessary. Now using the fact that B 2 T k, i.e.

·`…B† < ¯` for ` ¶ k, and the assumption ¯` ! 0 (` ! 1) we obtain the assertion

of the proposition. &

Observe that for every continuous function g there is constant C > 0 such that

dk µ C for all k. Thus for all " > 0 there is a K ˆ K…"† 2 N such that for all k ¶ K

219Approximation of attractors and invariant measures

Dow

nloa

ded

by [

Mac

quar

ie U

nive

rsity

] at

17:

53 2

8 O

ctob

er 2

014

…p…x; T dk† ·k…dx† µ C

…p…x; T† ·k…dx†

µ C

…p…x; T† ¸…dx† ‡ "

³ ´ˆ C":

Combining this with the fact that p…x; Tkdk† µ p…x; T dk† we conclude that

…p…x; Tk

dk† ·k…dx† ! 0

as k ! 1, ® nishing the proof for the transient part.

5.2. The supporting part S

In order to prove convergence on S we are going to use the fact that the diameter ofthe collections Sk is going to zero as k tends to in® nity. Furthermore, since

Sk‡1 » Sk, S can be viewed as the limit of the sets Sk, i.e. S ˆT1

kˆ0 Sk.

In the situation that we are interested in, the transition function p is absolutely

continuous. In this case it is easy to see that S has positive Lebesgue-measure.

Suppose this was not the case, i.e. m…S† ˆ 0. Then we had p…x; S† ˆ 0 for all x bythe absolute continuity of p and p…x; T† ˆ 0 for ¸-almost all x according to

Proposition 1. Since X ˆ S [ T this is a contradiction to the fact that p…x; ¢† is a

probability measure.

Proposition 2.…

Sk

dk dm ! 0 for k ! 1:

Proof. If we de® ne for a bounded measurable function g : X !

Qkg ˆX

B2Sk

B

m…B†

…

B

g dm;

we get…

Sk

dk dm ˆ…

jQBkg gj Sk

dm ˆ…

jQkg g Skj dm

µ…

jQkg g S j dm ‡…

jg S g Skj dm : …5†

For the second integral in (5) we get…

jg S g Skj dm µ max

x2Xjg…x†j m…SknS† ! 0

for k ! 1. The ® rst integral in (5) can be bounded as follows:…

jQkg g Sj dm µ…

jQkg Qk…g S†j dm

‡…

jQk…g S† g Sj dm: …6†

For the ® rst integral on the right hand side of (6) we get

220 O. Junge

Dow

nloa

ded

by [

Mac

quar

ie U

nive

rsity

] at

17:

53 2

8 O

ctob

er 2

014

…jQkg Qk…g S†j dm ˆ

… X

B2SkB\T 6ˆ1

B

m…B†

…

B

g g S dm

­­­­­­­

­­­­­­­dm

µ… X

B2SkB\T 6ˆ1

B

m…B†

…

BnS

jgj dm

Á !dm

ˆX

B2SkB\T 6ˆ1

…

BnS

jgj dm µ maxx2X

jg…x†j m…SknS† ! 0

for k ! 1. For the second integral on the right-hand side of (6) we use standard

arguments from measure theory (see, e.g. ManÄ e (1987)). We enlarge the collection Sk

to a partition SSk of X , requiring that diam SSknSk

± ²µ diam…Sk†. For gg 2 L1 we

consider the projection

QQkgg ˆX

B2SSk

B

m…B†

…

B

gg dm;

and observe that QQkgg ˆ Qkgg, if supp…gg† » Sk. We can now rewrite the second inte-gral on the right-hand side of (6) as

…jQk…g S† g Sj dm ˆ

…jQQk…g S† g S j dm:

Since diam…SSk† ! 0 we get kQQkgg ggk1 ! 0 for k ! 1, this ends the proof of

Proposition 2. &

Finally we use the last assumption of Theorem 1, namely that the transition

density h is bounded. We get…

p…x; Skdk† d·k ˆ

… …

Sk

dk…y† h…x; y† m…dy†³ ´

·k…dx†

µ K

…

Sk

dk dm ! 0

for k ! 1, according to Proposition 2.

To recapitulate, we showed that„

p…x; Tkdk† d·k ! 0 and

„p…x; Sk

dk† d·k ! 0

as k ! 1, i.e.…

p…x; dk† d·k ˆ…

p…x; jg QBkgj† d·k ! 0;

which according to the estimate (3) and (2) shows that the transition function pBkis

indeed a small random perturbation of p with respect to the family …·k†k. Using

Lemma 1 this ends the proof of Theorem 1.

AcknowledgementsThe author is grateful to Michael Dellnitz for helpful comments on the proof and to

Gary Froyland for thorough proofreading of the manuscript. He would also like to

thank the referees for valuable feedback.

221Approximation of attractors and invariant measures

Dow

nloa

ded

by [

Mac

quar

ie U

nive

rsity

] at

17:

53 2

8 O

ctob

er 2

014

ReferencesBlank, M., and Keller, G., 1998, Random perturbations of chaotic dynamical systems: stability of the

spectrum. Nonlinearity, 11(5): 1351± 1364.Boyarsky, A., Gora, P., and Lou, Y., 1994, Constructive approximations to the invariant densities of

higher dimensional transformations. Constructive Approximations, 10(1): 1± 13.Boyarsky, A., and Lou, Y., 1991, Approximating measures invariant under higher-dimensional chaotic

transformations. Journal of Approximation Theory, 65(2): 231± 244.Chiu, C., Du, Q., and Li, T., 1992, Error estimates of the Markov ® nite approximation of the Frobenius±

Perron operator. Nonlinear Analysis, 19(4): 291± 308.Dellnitz, M., and Junge, O., 1998, An adaptive subdivision technique for the approximation of attractors

and invariant measures. Computation and Visualization in Science, 1: 63± 68.Dellnitz, M., and Junge, O., 1999, On the approximation of complicated dynamical behavior. SIAM

Journal Numerical Analysis, 36(2): 491± 515.Ding, J., Du, Q., and Li, T. Y., 1993, High order apporoximation of the Frobenius± Perron operator.

Applied Mathematics and Computation, 53: 151± 171.Ding, J., and Zhou, A., 1996, Finite approximations of Frobenius± Perron. A solution of Ulam’s con-

jecture to multi-dimensional transformations. Physica D, 1± 2: 61± 68.Froyland, G., 1995, Finite approximation of Sinai± Bowen± Ruelle measures for Anosov systems in two

dimensions. Random and Computational Dynamics, 3(4): 251± 263.Froyland, G., 1996, Estimating physical invariant and space averages of dynamical systems indicators.

PhD thesis, University of Western Australia. Available from http://www.upb.de/math/Ä froyland.Guder, R., Dellnitz, M., and Kreuzer, E., 1997, An adaptive method for the approximation of the

generalized cell mapping. Chaos, Solitons and Fractals, 8(4): 525± 534.Hunt, F. 1996, Approximating the invariant measures of randomly perturbed dissipative maps. Journal of

Mathematical Analysis and Application, 198(2): 534± 551.Junge, O., 1999, Mengenorientierte Methoden zur numerischen analyse dynamischer Systeme. PhD thesis,

University of Paderborn. Available from http://www.upb.de/math/Ä junge.Keller, G., 1982, Stochastic stability in some chaotic dynamical systems. Montashefte fuÈ r Mathematik, 94:

313± 333.Khasminskii, R., 1963, The averaging principle for parabolic and elliptic diŒerential equations and

Markov processes with small diŒusion. Theory of Probability and its Applications, 6: 1± 21.Kifer, Y., 1986, General random perturbations of hyperbolic and expanding transformations. Journal

d’analyse mathematique, 47: 111± 150.Lasota, A., and Mackey, M., 1994, Chaos, Fractals and Noise (Berlin: Springer-Verlag).Lehoucq, R.B., Sorenson, D. C., and Yang, C., 1998, ARPACK Users’ Guide. Society of Industrial and

Applied Mathematics (SIAM), Philadelphia, PA. Solution of large-scale eigenvalue problems withimplicitly restarted Arnoldi methods.

Li, T.-Y., 1976, Finite approximation for the Frobenius± Perron operator. A solution to Ulam’s conjec-ture. Journal of Approximation Theory, 17: 177± 186.

ManÄ e , R., 1987, Ergodic Theory and DiVerentiable Dynamics (Berlin: Springer-Verlag).Miller, W., 1994, Stability and approximation of invariant measures for a class of nonexpanding trans-

formations. Nonlinear Analysis, 23(8): 1013± 1025.Murray, R., 1997, Discrete approximation of invariant densities. PhD thesis, University of Cambridge.Ulam, S. M., 1960, A Collection of Mathematical Problems (New York: Interscience).

222 O. Junge

Dow

nloa

ded

by [

Mac

quar

ie U

nive

rsity

] at

17:

53 2

8 O

ctob

er 2

014