File: 643J 253401 . By:BV . Date:15:02:96 . Time:16:20 LOP8M. V8.0. Page 01:01 Codes: 6475 Signs: 4316 . Length: 60 pic 11 pts, 257 mm oooia space topological n h maps the and 1.1. fractals. introduction and comprehensive systems a function for iterated the to [19] reviewing briefly See by work. start We classical algo- work. of this number on a based and rithms [25] 9]. Hutchinson 10, of work 30, theoretical 29, 11, and 6, nets, 1, 8, graphics, neural [7, computer automata, physics statistical as learning such found compression, has theory areas image The diverse 5]. 18, in 30, 29, applications and 13, seventies 2, late [25, in eighties nature early in [32] self-similarity Mandelbrot and of fractals work on seminal the since research of area 0890-5401 ] no. article computation and information 2534  Computation and Information be opeemti pc falnnepycmatsbesof subsets compact non-empty X all of space metric complete as n a siaeteitga pt n hehl faccuracy. of threshold any to with up integral functions the Lipschitz estimate and continuous can IFSs recurrent one everywhere hyperbolic the maps, For for almost distribution. formula this to a of respect provide to value used is screen. expected integral digitised the Riemann on generalised distribution the The invariant of an the obtain uniqueness generate and to IFS and recurrent algorithm hyperbolic existence weakly the a of prove distribution invariant then for algorithms We attractor corresponding IFSs. the the hyperbolic to extending for measure, approximations invariant digitised Hutchinson the and and algo- of discrete finite obtain present results also to We IFS setting. rithms classic general weakly hyperbolic more this the a weakly in IFSs a of hyperbolic extending of attractor measure probabilities, the invariant of with the uniqueness and and IFS existence hyperbolic the the of uses respectively, prove domain that which power we generalises probabilistic model, the which domain-theoretic and domain a space, power Plotkin on metric Based compact IFS. a hyperbolic on (IFS) system l ihso erdcini n omreserved. form any in reproduction of rights All Copyright maps eitouetento fwal yeblcieae function iterated hyperbolic weakly of notion the introduce We 96Aaei rs,Inc. Press, Academic 1996 An nti ae,w ilb anycnendwt h basic the with concerned mainly be will we paper, this In active an bee has system function iterated of theory The ihthe with hyperbolic trtdFnto Systems Function Iterated f trtdfnto system function iterated i  :  96 X  96b cdmcPes Inc. Press, Academic by 1996 0014 d 18.00 Ä Hausdorff H ( o opeemti space metric complete a For . X A f , i ( B r l otatn,te h F ssi to said is IFS the then contracting, all are i 1 ..., =1; )=inf X .INTRODUCTION 1. oe oan n trtdFnto Systems Function Iterated and Domains Power sgvnb iiesto continuous of set finite a by given is metric [$ N 124 | .If ). B eateto Computing of Department d 89 (1996) 182197 , H ( X IFS A eie by defined sacmlt ercspace metric complete a is $ ) and [X 180 ; A f 1 Queen X , f B let , 2 $ ..., , ] ' Gate s , , H meilCleeo Science of College Imperial f X -al ae E-mail: N ] , ba Edalat Abbas ethe be odnSW London na on Ä 182 doc.ic.ac.uk eie by defined sthe is locnrcigwt otatvt factor contractivity with contracting also s a is bandb Hepting by attractor obtained the to first approximation was discrete a generate to rithm r contracting are set subset compact non-empty a for where, hn the Then, IFS hc scle the called is which theorem, mapping the called is hteach that oordiae ncmue rpis yeblcIFS hyperbolic A graphics. probabilities computer with in images result, a coloured as and, distributions probability invariant produces 1.2. 2.3. Section in described is It 33].) mohlclsrcue oefr fsl-iiaiy and, self-similarity, of non-integral form a some usually, structure, local smooth a usually is ,2] ti sue that assumed is it 21], 6, a called also is tor subset compact non-empty any for i 72 stecnrciiyfco of factor contractivity the is yeblcISidcsamap a induces IFS hyperbolic A hr sas rbblsi eso fteter that theory the of version probabilistic a also is There o plctosi rpisadiaecmrsin[1, compression image and graphics in applications For BZ [X F ihProbabilities with IFS -aallbody $-parallel , ntdKingdom United ; f , 1 f akvoperator Markov ehooyadMedicine and Technology i , F (1 f ( fractal 2 A C 0< contractivity ..., , )= $ = affine i p  f i [x f N [X efsmlrset self-similar attractor 1and <1 .. thsfn,cmlctdadnon- and complicated fine, has it i.e., ; F 1 N ] ( A A of with , T # ; a nqefxdpoint fixed unique a has sasge probability a assigned is ) rnfrain.Te,teattractor the Then, transformations. F =lim *= ) f : X asof dimension Hausdorff C 1 _ : M X ..., , | H . _ n f steplane the is 1 fteIS ytecontracting the By IFS. the of X y Ä X 2 sdfndby defined is X fteIS n ehave we and IFS, the of ( # f A Ä Ä opc ercsae such space, metric compact a N , C f ) ; i F tal et H _ M p } (1 . n d 1 X ( 1 ( ..., , }}} A A ,(1) X x i : =1 2] Seas [14, also (See [24]. . N )(2) , i R _ y  p C X p ) 2 N i f N s n httemaps the that and =1. ] N 2] h attrac- The [25]. =max .Tenumber The ). $] ( X sahyperbolic a is A iiealgo- finite A . and .I fact, In ). A p i i *in $ s with 0 the 0, i where ; H F (3) X is s , File: 643J 253402 . By:BV . Date:15:02:96 . Time:16:20 LOP8M. V8.0. Page 01:01 Codes: 6436 Signs: 4861 . Length: 56 pic 0 pts, 236 mm hs upr steatatrof attractor the is support whose point fractal aig( making oevr o l otnosfunctions continuous all for Moreover, h olwn ovrec ihpoaiiyone, probability with convergence following the o n oe subset Borel any for ucisnmti r metric Hutchinson i hc ie h xetdvleof value expected the gives which h iceetplg.Choose topology. discrete the osrc h sequence the construct hyperbolic, h au fteitga facniuu ucinwith function respectively. 5, are continuous they and 3.3 [24]; a Sections in obtained in of also described discrete were integral measure this a the to respect for of generate formula a value to and the measure invariant algorithm this to attractor finite approximation the attractor, colour A can the one of accordingly. scheme, regions colouring a different using in and densities point ferent rbblte 1,1 sbsdo h olwn ergodic following the on Let based [18]. Elton is of theorem 1] [13, probabilities hoe,i ssonta h ek oooyadthe and topology weak* on the topology that metric shown Hutchinson is Alaoglu's it including theory, theorem, space Banach some using Then, ie by given F ihpoaiiiso h opc ercspace metric compact the let on probabilities with IFS nteset the on measure e of set oobtain to ie,frams l sequences all almost for (i.e., oe measure Borel a esr fteISand IFS the of measure L & ( scoe ihprobability with chosen is The n B x , hn lo' hoe asta,wt rbblt one probability with that, says Theorem Elton's Then, . r B 0 H + X etenme fpit nteset the in points of number the be ) When . # ( *of + X admieainalgorithm iteration random uhthat such + , M M sas alda called also is * & eayiiilpit Put point. initial any be i )=sup 1 T 2 T 1 X X ilb otatn a.Teuiu fixed unique The map. contracting a be will , and X hndfnsapoaiiydsrbto on distribution probability a defines then r fnraie oe esrson measures Borel normalised of T | H + ( opc ercsae fteISis IFS the If space. metric compact a ) gd+ + + R + { # H x | )( *( *( | n f M 2 a edfndon defined be can X hsivratdsrbto ie dif- gives distribution invariant this , B ( = B =lim *= # x (x )= 1 fd+ ( =lim )= B X )& B f i n 2 n # )0 where ))=0, ( oaBrlmeasure Borel a to i ) Ä n : ( =1 & x N f efsmlrmeasure similar self X Ä B [X n ( 1 (x 0 When . stebudr of boundary the is ) | )= y p i p X 1 )|  ; i i n Suppose . L + OE OAN N TRTDFNTO SYSTEMS FUNCTION ITERATED AND DOMAINS POWER Let . fd& # f ) f n i g 1 ( ( =0 [X n i 7 n n . n ..., , +1 M 2 f 0 d ( +1 , N i &1 ( | f ; g B 1 x f i 1 7 X trno uhthat such random at ( f ( : f # , M ( ) X x x o nISwith IFS an for 1 x N . B N X y + g 1 7 ..., , i 0 ; onie thereby coincide, ) ), steinvariant the is * 1 = )) = : ) nti way, this In )). Ä ,(4) X scmat the compact, is | N p B X \ 1 ,w have we ), [ [x f sflos[2]: follows as f x saBrlsub- Borel a is R ..., , Ä i ,..., 1, N 1 , ( , 0 ] x y T R , ra or 0 p X ( # 2] The [25]. x ehave we , .Repeat ). + N N] ttakes It . 1 X )# ] ..., , = B X multi- ean be . M Let . with and x 1 n X X ] eoe let before, imtro n set any of diameter eurn F.Frahproi eurn F,cnie a consider IFS, recurrent on hyperbolic walk random a For IFS. recurrent uhthat such let x inlpoaiiythat probability tional a rs o xml nacmatinterval map compact smooth a a in contains IFS example one where the situation for 17], This if hyperbolic. 9, arise not [23, are can applica- nets which of neural IFSs number in encounters a example by for motivated tions, is definition Our IFS. power probabilistic fixed space. the the upper on the is function of space domain continuous metric a compact of IFS a point hyperbolic on a probabilities of measure with upper invariant the con- the of a domain Similarly, of power space. point Plotkin fixed the unique compact on a the function tinuous as on obtained IFS is hyperbolic a space of metric attractor was the It that probabilities. with shown IFSs and IFSs for models theoretic 1.4. image [6]. IFSs interactive recurrent hyperbolic an on based is is system, which modelling Recurrent fractal System), (Vector for Function VRIFS system Iterated Barnsley's software compression, practical image first iteration random The the to algorithm. reduces above walk measure random invariant the the and just then is distribution stationary the of ent n otnet bantesequence the obtain to Continue rbblte sflos[] Let and [4]. space follows as probabilities 1.3. n trigcode starting a and p oi eurn F.Ti eeaie h hoyo hyper- if a of fact, theory In to probabilities. the with generalises one IFSs This bolic probability IFS. recurrent with bolic converges on sequence measure this of nindecomposable an | f i 1 ihtepouttopology. product the with 1) 1 0 $( j e ( Let ewl eeitouetento fawal hyperbolic weakly a of notion the introduce here will We domain- construct to used were domains power [15], In eurn trtdfnto ytm eeaieIS with IFSs generalise systems function iterated Recurrent Then = v v v 7 stecniinlpoaiiythat probability conditional the is x ekyHproi IFS Hyperbolic Weakly eurn IFS Recurrent | N f |1btnt| not but )|1 o all for p  i i 1 ij etesto l niiesequences infinite all of set the be ( N j 0frall for 0 hnw banahproi F ihprobabilities; with IFS hyperbolic a obtain we then X x =1 [X 0 , p .Te pick Then ). [X 7 d i p ; 1 X eacmatmti pc;w eoethe denote we space; metric compact a be ) i N i f ij 2 , j ; 1frall for =1 p = ; aldthe called j f i p 2 1 hr exist there i [ 3 ij X , ,2 ..., 2, 1, }}} ; f i i 2 sflos pcf trigpoint starting a Specify follows. as , a i , N ..., , 0 p f j j and , # $( 1 ,..., 2, =1, i _ n &1 j x 7 X i f N ttoaydistribution stationary scoe,adput and chosen, is i )|<1. i 2 N N] N n , y| by >0. # ] ikanumber a Pick . o-tcatcmti,i.e., matrix, row-stochastic i 7 1 hproi)IS e ( Let IFS. (hyperbolic) a ihtedsrt oooyand topology discrete the with , a N i |=sup 2 N] uhthat such ..., , (x X n scle (hyperbolic) a called is i ) n j eacmatmetric compact a be [d n with scoe,addefine and chosen, is 0 f p i ( : ij 1 x h distribution The . X = i p , i 2 x i 1 y i 1 i Ä p 1 3 # 2 j )| = j = ...( stecondi- the is 7 ftehyper- the of X sindepend- is x i N , f i i n and y satisfying 2 uhthat such ( # # f i 7 a] x 1 p X ( i ij N 0 x n )be .As 183 # 0 = for )). R X j File: 643J 253403 . By:BV . Date:15:02:96 . Time:16:20 LOP8M. V8.0. Page 01:01 Codes: 6733 Signs: 5374 . Length: 56 pic 0 pts, 236 mm f tflosta lim that follows It eaieitgr ih0 with integers negative 184 n yeblcIFS hyperbolic any lal o otatn naeae u ti vnulycon- eventually is it as but tracting average, on contracting not clearly o all for lal yeblcISi ekyhproi.Oesimilarly a One hyperbolic. weakly defines is IFS hyperbolic a clearly f n otatn.Ti a ese vnfrtecs fasingle a of case the for ( even eventually seen be map be not hyperbolic. can need weakly This IFS contracting. is hyperbolic weakly IFS a contracting However, eventually an that adt be to said ie on at point fixed lim g ing map hyperbolic h xeddIFS extended the j sadcesn euneo ust of i subsets of sequence decreasing a is fore, a,atiedfeetal a with map differentiable twice a say, and eeaieeetal otatn IFSs. IFSs contracting hyperbolic eventually not generalise weakly is but Therefore, 2.6) contracting. Proposition eventually (see hyperbolic weakly is above, [ i rcigo vrg.Ti a ese vni h rva IFS trivial the in even seen be [ can con- This be average. not on need tracting probabilities) with IFS hyperbolic weakly a ekyhproi F ntecs facmatmti space metric compact a of of X case notion the the in with IFS these hyperbolic compare weakly We literature. the in maps IFS recurrent hyperbolic weakly ulycnrcigISi ekyhproi sflos We follows. as any hyperbolic write weakly can is IFS contracting tually that | 1 p ( 0 f i m 1 tflosb h envletermthat theorem value mean the by follows it 1, 0 n 0 1], [0, 0 1]; [0, nIFS An . ( 1 , i # i ekyhproi Fsgnrls yeblcIS since IFSs hyperbolic generalise IFSs hyperbolic Weakly Definition nISwt probabilities with IFS An hr r w te oin fIS ihnon-contracting with IFSs of notions other two are There :[0,1] &1) 1 x 2 i n f }}} 2 7 =ah( )=tanh 2]i hr ssome is there if [21] Ä i ..., , 2 f f [ i k k N 2 }}} k " 0 +2 0 1], [0, x N = :[0,1] ( ecnld htlim that conclude we , x , | 1 on =1) i f f f y < o all for )<0 k f f 0 i 1 Ä n ekyhproi F ihprobabilities with IFS hyperbolic weakly }}} [X i i ] ] otatn naverage on contracting # 1 1 ffralifnt sequences infinite all for if # X f f f swal yeblc ic ( Since hyperbolic. weakly is 1 2 X 7 0 ]i o otatn o any for contracting not is 1] [0, i || i with i 2 ( 2 ; x x mk f neetal otatn F ad hence, (and, IFS contracting eventually An . x }}} }}} N k Ä 1 f n i ) Then, )). 0 ` =1 0adlim and =0 1 N = o all for )=0 . ] 1as 1 1 , flength of o 1 for f f 0 ]with 1] [0, . f snteetal otatn.I at for fact, In contracting. eventually not is n i i g d n k 2 X f Ä [ ( X j ..., , 1 [ 1 r otatn o l iiesequences finite all for contracting are nIFS An [,1.Let 1]. =[0, ( 0 1]; [0, f 0 1]; [0, g |=0. x i j ( 2 =a(,(5 )=max(0, x x | n f }}} m f f ), = 0 ](e.g., 1] [0, # N q 0 0 n  ]  k [,1)= n,hne h IFS the hence, and, 1])|=0 ([0, g f pk a nqewal attracting weakly unique a has n x ti ayt e hta even- an that see to easy is It . i ssi obe to said is k j Ä f ( p p f k f 1 0 ] ecnas d the add also can We 1]. [0, # 1sc htthe that such 1 X 0 y + .As 2 1 Since &1. ..., , [X , ( )) n . where | [X x f q Ä f p 1 )=(1& where , ; i n ..., , 3 fteeis there if [3]  g f ; ( f N x j 1 x f f p sd | , ] 0 1 = o all for )=0 scnrcigfrany for contracting is f f tcnb hw that shown be can it , )2.Ti F is IFS This 2)&2). :[0,1] ..., , f i ( i 2 1 N 1 x f f vnulycontract- eventually ..., , f (f ] 0 i 0 X , i x f (0)=0, 2 2 ( p where , h f x ...# tflosthat follows it , }}} 0 n ) N i j )  1 $()1frall for (0)=1 )$ and f )= m ooti the obtain to 2 f ; N f i Ä p 2 = n ] i 7 n }}} 1 x 1 There- 1. X ..., , q 0 ]be, 1] [0, | N (1& x s is i N = and |=0 f ( 1such <1 f r non- are m #[0,1]. i have we f n $ 0 &1) k 0 X) p weakly (0)=1 BA EDALAT ABBAS n a and x N maps sas is k )or ] +1 n 0 is aiysont eeetal otatn aycomposition (any contracting f eventually be to shown easily F edntb vnulycnrcig.Ti a ese by seen be hence, can probabilities (and, This with hyperbolic contracting). IFS weakly eventually the be be not not contracting need is need which IFS average an hand, on other the On average. on oes(oe=atal ree set). ordered (poset=partially posets 1.5. uniformly integrated all model. are domain-theoretic the measures, within respect invariant everywhere with these functions) almost to Lipschitz of (or value functions expected measures, continuous the invariant of the computation to and with algorithms IFS attractor finite of the hyperbolic that the (b) weakly and approximate IFS, IFS a recurrent hyperbolic of or uniqueness probabilities weakly measure and a existence invariant of of the theory proof attractor the the the of (a) of aspects namely several IFS, that of feature unifying the has hyperbolic a of distribution accuracy, invariant IFS. of the recurrent threshold to given of any value respect to expected with up the for map, expression Lipschitz simple any a distribu- of this also to value and respect expected tion with the function for continuous formula digitised almost a a an deduce on algo- also distribution We finite invariant screen. a this obtain generate a to and of rithm IFS distribution recurrent invariant hyperbolic the the weakly prove of then weakly uniqueness will and and We existence IFSs probabilities. with hyperbolic IFSs of weakly hyperbolic results to the Hepting above extend mentioned of to different the those will model we of Hutchinson, a paper, domain-theoretic uniqueness this In the needs and setting. use general existence more one this the in attractor contracting, prove to necessarily approach not is dif- totally IFSs. a hyperbolic with represent compared average as class on ferent contracting are which F ihprobabilities with IFS all f nqeatatr nfc,tecmatsbes[,1 and 1] [0, subsets compact the does fact, IFS In this attractor. that unique Note a hyperbolic. weakly not is n,hne h F scnrcigo vrg.Hwvr for However, average. on contracting all is IFS the hence, and, c h opeiyaaye fteeagrtm,ad()the (d) and algorithms, these of analyses complexity the (c) [ 1 i 0 1 ( ( A ercl h ai eiiin nteter fcontinuous of theory the in definitions basic the recall We show, will we IFS, of framework domain-theoretic The map the IFS, hyperbolic weakly a for Since f ] x n x i 2 )= )= 1w have we 1 , oainadTerminology and Notation f r ohfxdpit of points fixed both are y i 3 0 ]w have we 1] [0, # x a otatvt 5 contractivity has f 1 | and 3 ( f A 1 ( ) x _ )& f f 2 2 f f ( ( 2 n A 1 x [,1)[,1 n,teeoe h IFS the therefore, and, 1] 1])=[0, ([0, ( )=min(2 .W hrfr ocueta IFSs that conclude therefore We ). y || )| [ f 2 0 1]; [0,  ( )bti gi o contracting not again is but 8) [ x 0 1]; [0, )& x ) ti aiyse htfor that seen easily is It 1). , F f tal et : 2 f H ( 1 y , 0 1] [0, f )| f 1 n hs n[15] in those and . 2 , ;1 f 2  3 2 ,1 2, ;1 Ä | x  & ,1 2,  H F 2 y 0 ]with 1] [0, ] : | H  2 2 hc is which not X ] Ä where have H X File: 643J 253404 . By:BV . Date:15:02:96 . Time:16:20 LOP8M. V8.0. Page 01:01 Codes: 6165 Signs: 4617 . Length: 56 pic 0 pts, 236 mm b f of image of image the venient, subset any and ( rdc oooycicd.A ( An coincide. topology product onal ai.Tepouto ( ( of product The basis. countable D pe on lb fadrce e fcmatsbesis subsets if compact relation only of way-below and set The directed intersection. a of their (lub) namely bound element, upper bottom a with dcpo lmnsof elements rpriso h pe pc,freape rm[5.The [15]. from ( example, order for partial space, upper the of properties X hc suwr lsd(i.e., closed upward is which lub. a has subset bounded by denoted subset directed every which ietdset directed o n aro elements of pair any for subset a rsre uso ietdst,i.e., sets, directed of lubs preserves ( 2.1. IFSs. studying for a-eo y way-below A oe measures Borel where olw htacniuu function continuous a that follows ercspace metric space topological ncesbeb uso ietdst (i.e., sets directed of lubs by inaccessible ihlateeet(rbottom) by (or given element least with ct oooyifi is it iff topology Scott dcpo x |- |- U &1 R # ietdcmlt ata order partial complete directed o n map any For noe set open An Let esatb rsnigtedmi-hoei framework domain-theoretic the presenting by start We subset non-empty A ie w elements two Given is X ree yrvreicuin ercl h following the recall We inclusion. reverse by ordered O cniuu cowt cutbe basis (countable) a with dcpo )continuous cniuu coi hc h ct oooyadthe and topology Scott the which in dcpo )continuous B b , h pe Space Upper The continuous .I a esonta function a that shown be can It ). o all for D $ nta of instead X [x )of oaohrone another to B  A eacmatHudrfsae The space. Hausdorff compact a be i | .ADMI-HOEI MODEL DOMAIN-THEORETIC A 2. n i by B 0 eoe by denoted , A X b # B D  X hnteeis there then , # I] way-below fA ossso l o-mt opc ust of subsets compact non-empty all of consists O U ednt by denote we , otisanihorodof neighbourhood a contains f sa is B A f fi a ai;i is it basis; a has it if + X n : . oe is poset A . B saydrce ustof subset directed any is ( nta of instead on D , f = X D $ basis &1 Ä ). X sdntdby denoted is fthe of E sabuddcmlt continuous complete bounded a is ) ( x monotone E A B ednt,weee oecon- more whenever denote, we , with E x by n point any , .Teltieo pnst fa of sets open of lattice The ). x for , x d scniuu ihrsett the to respect with continuous is , y R P sdrce and directed is A y Scott fx f # a D + x ( y fapst( poset a of a es pe on (lub), bound upper least a has A A ( nadcpo a in # M fwhenever if , # nta of instead X ffreach for if n h r-mg of pre-image the and ) one complete bounded A i.e., , hr is there O |- OE OAN N TRTDFNTO SYSTEMS FUNCTION ITERATED AND DOMAINS POWER )= c ( = oooyo coi set a is dcpo a of topology X dcpo with |- 6 cniuu cosi an is dcpo's )continuous ,0  x a a has f |-continuous c 0 ) x : x . # i algebraic X # ( D saprilodrin order partial a is ) = = C D C X I x z nwihteleast the which in ,  c P Ä n subset any , f = # .Fracompact a For ). C f 1 h e fall of set the 1, d ( , y D y f A d ( es ie point fixed least A x = C D D : # x ( O = esay we , i B x a y D # )= ,teforward the ), D rmti it this From . with esythat say We . ) C nadcpo a on )is = R C O  y Ä = C pe space upper B h Scott The . h set the coi an is dcpo # f O C A  directed E ( satisfying O fi a a has it if x f  esay We . fevery if _ od if holds , ( n is and ) A rma from x i y y A # and ) = I # o a for C B A x x A by i of D D is z ), if } . ossigo iieuin fcoue frltvl compact of relatively of sets closures of open unions finite of consisting ipyas simply U map continuous embeds nue ct-otnosmap Scott-continuous a induces g on topology ete tteorigin the at centred nodrt omlt utbeter fISwith IFS of theory suitable a that assume to formulate need we to probabilities, and point order fixed this of in uniqueness the on result satisfactory a ehave we F okwt h IFS in the lie with then work will IFS the of of interior ercsaeand space metric order- the Since way. same in the exactly ing in defined are they as ustof subset iiyfco of factor tivity ihteEcienmti and metric Euclidean the with xssannepyrglrcmatset compact regular non-empty a exists opc asof pc X space Hausdorff compact a mt opc ust)btdfeettopologies. different but subsets) compact empty F sSotcniuu n a hrfr es ie point fixed least a therefore has namely and Scott-continuous is ( ( U U ( : f 1 A a nteohrhn if hand other the On aah sdteuprsaet oetefloigresult. following Proposition the note to space upper the used Hayashi o ovnec,w s h aentto o h map the for notation same the use we convenience, For X X H ( eua lsdsti n hc seult h lsr fisinterior. its of closure the to equal is which one is set closed regular A = C )= , , X $ $ )= [C Ä U , f X n ( and ) )isan 1 X F f ( H # X noteset the onto A ( ( F f A U C A X .If srvreinclusion, reverse is with ) : se[5 em .0) h nqeattractor unique The 3.10]). Lemma [15, (see X ) ;t eptenttossml ewl write will we simple notations the keep to ); _ U si q 1 n h map the and (1) Eq. in as U |- X A | H X X f A f ..[22] 2.1. C X oeta h w oooia spaces topological two the that Note . i A 2 otnosdp n a onal basis countable a has and dcpo continuous [X X [ F Ä *= ( (1 f where , si atacmatmti pc,then space, metric compact a fact in is A ( R , : a ai ie ytecollections the by given basis a has A [A ; U d ) f X a] max O 1 *)= f _ H ' X s ( n 1 Ä i ; aetesm lmns(h non- (the elements same the have ) ( A ..., ,  with ( X s f }}} F a ) 1 : X i Y fmxmleeet of elements maximal of ) N _ n # A ..., , X x A A ( f[X If _ f 0 salclycmat complete compact, locally a is .Hwvr nodrt obtain to order in However, *. X ,te ti ayt hc that check to easy is it then ), d fcmatHudrfspaces Hausdorff compact of [ Ä N f , saycoe alo radius of ball closed any is n,teeoe ecnsimply can we therefore, and, ( 2 ( )= ] f hntemap the then O ( f X N U A [x] s 1& N yeblcIS hnthere then IFS, hyperbolic a , i ( ; A ) h igeo map singleton The )). ] ,0 ) X A U , f npriua,if particular, In . f n _ stelretcompact largest the is * i 1 s ) ( f , O i F : }}} f U X )) n s 2 A ( F i ..., , X , 1 stecontrac- the is <1, X ,where %, _ samti space. metric a is : Ä U ). f f 1 X N U N ( Ä sa F on IFS an is ] A A Y ) U eie by defined uhthat such A U X sthe is % X X above, Any . is 185 U R R n f , 186 ABBAS EDALAT

FIG. 1. The IFS tree. where d is the Euclidean metric. Therefore, as far as a hyper- Proof. The implications (i)  (ii) and also (iii) O (i) are bolic IFS on a locally compact, complete metric space is all straightforward. It remains to show (i) O (iii). Assume concerned, there is no loss of generality if we assume that that the IFS does not satisfy (iii). Then there exists =>0 the underlying space X is a compact metric space. We will such that for all n0 there is a node on level n of the IFS make this assumption from now on. tree with diameter at least =. Since the parent of any such

Let X be a compact metric space and let [X; f1, ..., fN] be node will also have diameter at least =, we obtain a finitely an IFS. The IFS generates a finitely branching tree as in branching infinite subtree all whose nodes have diameter Fig. 1, which we call the IFS tree. Note that each node is a at least =.ByKo nig's lemma this subtree will have an subset of its parent node and therefore the diameters of the infinite branch (fi1 fi2 }}}fin X)n0. Therefore, the sequence nodes decrease along each infinite branch of the tree. The ( | fi1 fi2 }}}finX|)n0 does not converge to zero as n Ä IFS tree plays a fundamental role in the domain-theoretic and the IFS is not weakly hyperbolic. K framework for IFSs: As we will see, all the results in this Corollary paper are based on various properties of this tree; these 2.3. If the IFS is weakly hyperbolic, then for any sequence i i }}} #7|, the sequence ( f f }}}f x) include the existence and uniqueness of the attractor of a 1 2 N i1 i2 in n0 weakly hyperbolic IFS, the algorithm to obtain a discrete converges for any x # X and the limit is independent of x. approximation to the attractor, the existence and unique- Moreover, the mapping ness of the invariant measure of a weakly hyperbolic IFS: with probabilities, the algorithm to generate this measure | ?: 7N Ä X on a digitised screen, the corresponding results for the recurrent IFSs, and the formula for the expected value of an i1 i2 }}} [ lim fi1 fi2 }}}fin x almost everywhere continuous function with respect to the n Ä invariant distribution of a weakly hyperbolic recurrent IFS. n We will now use this tree to obtain some equivalent is continuous and its image is A*=n0 F X. characterizations of a weakly hyperbolic (IFS) as defined in An IFS [X; f , ..., f ] also generates another finitely Definition 1.1. 1 N branching tree as in Fig. 2, which we call the action tree. Here, a child of a node is the image of the node under the Proposition 2.2. For an IFS [X; f , ..., f ] on a com- 1 N action of some f . pact metric space X, the following are equivalent: i (i) The IFS is weakly hyperbolic. Note that the IFS tree and the action tree have the same (ii) For each infinite sequence i i }}} #7|,the intersec- 1 2 N set of nodes on any level n0. tion n1 fi1 fi2 }}}finX is a singleton set. (iii) For all =>0, there exists n0 such that Corollary 2.4. If the IFS is weakly hyperbolic, n | fi1 fi2 }}}finX|<= for all finite sequences i1 i2 }}}in #7N of limn Ä | fin fin&1 }}}fi1X|=0 for all infinite sequences | length n. i1 i2 }}} #7N.

File: 643J 253405 . By:MC . Date:12:02:96 . Time:08:52 LOP8M. V8.0. Page 01:01 Codes: 4431 Signs: 3070 . Length: 56 pic 0 pts, 236 mm POWER DOMAINS AND ITERATED FUNCTION SYSTEMS 187

FIG. 2. The action tree.

Conversely, we have the following. Recall that a map all infinite branches of this tree. Such a set is an example of f: X Ä X is non-expansive if d( f(x), f( y))d(x, y) for all a finitely generable subset of the |-continuous dcpo UX as x, y # X. it is obtained from a finitely branching tree of elements of UX. This gives us the motivation to study the Plotkin power Proposition . . 2 5 If each mapping fi in an IFS is non- domain of UX which can be presented precisely by the set of expansive and limn Ä | fin fin&1...fi1X|=0 for all infinite finitely generable subsets of UX. We will then use the | sequences i1 i2 ...#7N, then the IFS is weakly hyperbolic. Plotkin power domain to prove the uniqueness of the fixed point of a weakly hyperbolic IFS and deduce its other Proof. Assume that the IFS is not weakly hyperbolic. properties. Then, by condition (iii) of Proposition 2.2, there exists

=>0 such that for each n0 there is a node fin fin&1 }}}fi1 X 2.2. Finitely Generable Sets on level n of the action tree with diameter at least =. Since, The following construction of the Plotkin power domain by assumption, fi is non-expansive, it follows that the n of an |-continuous dcpo and the subsequent properties are parent node fin&1 }}}fi1 X has diameter at least =. We then have a finitely branching infinite subtree with nodes of a straightforward generalization of those for an |-algebraic diameter at least =. Therefore, by Ko nig's lemma, the action cpo presented in [35, 36]. Suppose (D, =C ) is any |-con- tree has an infinite branch with nodes of diameter at least =, tinuous dcpo with bottom and B D a countable basis for which gives a contradiction. K it. Consider any finitely branching tree, whose branches are all infinite and whose nodes are elements of D and each Proposition child y of any parent node x satisfies x C y. The set of lubs 2.6. If [X; f1 , ..., fN] is a weakly hyperbolic = of all branches of the tree is called a finitely generable subset IFS with non-expansive maps fi : X Ä X (1iN) and if of D. It can be shown that any finitely generable subset of D [X; fN+1, ..., fM] is a hyperbolic IFS, then [X; f1 , ..., fN , can also be generated in the above way by a finitely branch- fN+1, ..., fM] is a weakly hyperbolic IFS. ing tree of elements of the basis B, such that each node is | way-below its parents. We denote the set of finitely Proof. Let i1 i2 }}} #7M. If the set [n1|N+1 generable subsets of D by F(D). It is easily seen that inM] is infinite, then clearly limn Ä | fi1 fi2 }}}finX|=0. If, on the other hand, the above set is finite, then it Pf (B) Pf(D) F(D), where Pf (S) denotes the set of all has a maximum element m1 say. Hence for all n>m we finite non-empty subsets of the set S. For A # Pf (B) and C # F(D), the order REM is defined by A REM C iff have N+1inM and, therefore, | fi1 fi2 }}}fim }}}finX|

|fim+1 }}}finX| which tends to zero as n Ä . K \a # A_c # C } aRc and \c # C_a # A } aRc. By Proposition 2.1, we already know that a weakly hyperbolic IFS has a fixed point given by A*=n0 n n n F X=n0 F X. Note that F X is the union of the nodes This induces a pre-order on F(D) by defining C1 =C EM C2 of the IFS tree on level n, and that A* is the set of lubs of iff for all A # Pf (B) whenever A REM C1 holds we have

File: 643J 253406 . By:MC . Date:12:02:96 . Time:08:53 LOP8M. V8.0. Page 01:01 Codes: 4574 Signs: 3234 . Length: 56 pic 0 pts, 236 mm File: 643J 253407 . By:BV . Date:15:02:96 . Time:16:20 LOP8M. V8.0. Page 01:01 Codes: 6368 Signs: 3829 . Length: 56 pic 0 pts, 236 mm ( by level at nodes of set The sbfr n osdrteSotcniuu map Scott-continuous the consider and f before as each of point fixed least of point fixed note we Finally, only. dcpo itself any for of that, consists class equivalence oss of consist omts hc a eesl enb osdrn the considering by seen basis easily the be to restriction can which commutes, h oooemap monotone the coecp that except dcpo pc and space hc,frcneine ednt by denote map we Scott-continuous convenience, for a which, to extension unique a rsnaino h lti oe oani em of terms in domain diagram power The subsets. Plotkin generable finitely the our of context sake of the the in for presentation here them [15]; reiterate we in completeness, shown of were properties following The A oooemap monotone U utet( quotient domain power convex C re.Abssi ie y( by given is basis A order. ai y( by basis a lmn f( of element mal lmnso one complete bounded a of elements 188 $ F : 2 : CU R o let Now = CU C ( f on D EM n [A n X EM [X] ), 0, X Ä C = F j C Ä |1 C U 2 F eas osdrteSotcniuu map Scott-continuous the consider also We . P CU ( EM D hn( Then . 1 f A U D U f n .If : [X ( ( ' 0 be f B P D sgvnby given is ) X a otmeeet namely element, bottom a has ) j f X ny If only.  n f ), ; [A eie nteaoebssb the by basis above the on defined , ) [X] oafxdpitof point fixed a to ( f U  F U f = hc sdfndo h basis the on defined is which $ C M] A 1 n = C X [X] X j , ..., , F E ( |1 = EM # C f D = ) U EM where n oooemap monotone any , [ U U F ( Ä otelatfxdpitof point fixed least the to Ff ), D EM : .The ). f F C = X X P D N P = sapeodrrte hnapartial a than rather pre-order a is C ( j [f CU ), n n  D  f ] D f U ww Âw w Âw ( n $ ( fteISte ste represented then is tree IFS the of ' EM 0 U a otmelement bottom a has = U i C and ) ,weeteeuvlnerelation equivalence the where ), X [X] M] ( nIS Let IFS. an of X U U A X X P n t qiaec ls will class equivalence its and ) s sbfr,acmatmetric compact a before, as is, C EM U lti oe domain power Plotkin f j .I olw that follows It ). ) D )|1 ( 1 [ f Ä D = eoe an becomes ) $ n CU CU ste eie ob the be to defined then is D [X] ), 1 CU F F A C then , = C oevr tmp the maps it Moreover, . . n X X j j 2  X  ossso maximal of consists X EM g = N n,therefore, and, , iff . M n n countable a and ) A ' 0 A ,1 g F C j . : : 1 ilb maxi- a be will F P U = C |- f n [ ( X U i X D  = F continuous EM P g . Ä ic for since , asany maps f ) : ] N] ( Ä = C n its and , U C U D rthe or then , BA EDALAT ABBAS X 2 . E X Ä )by and has be E ti ayt e that see to easy is It of F olwn em hs tagtowr ro somitted. is proof straightforward whose lemma following U iieybacigte uhta h e fndsa level at nodes 1 of than set less the that such n tree branching finitely iga commutes: Further- of diagram element. Scott-continuity maximal a the is it more, remark, above the by and, Therefore, h trco fteIFS the of attractor the bolic tflosta h es ie on of point fixed least the that  follows It of However, atsbe ftemti pc X space metric the of subset pact nteohrhn,for hand, other the On and hyperbolic es nfc ehave singleton of we consists fact set this in hyperbolic, sets; weakly is IFS the for ie on of point of fixed point fixed unique since the indeed is f is atflosimdaeyfo rpsto . and 2.7 of Proposition from points immediately fixed follows of part set first the precisely is [D : 0cnit ftecoueo pnsbeswt diameters with subsets open of closure the of consists 0 n X n CU Proposition nodrt e h eeaiaino q 2,w edthe need Lemma we (2), Eq. of generalization the get to order In Proof Proof Theorem f F A 0 i oeas that also Note . 1 hscnb hw o xml ycntutn a constructing by example for shown be can This . . |1 S Ä , f A hntempF map the then S X f n K A i * n [X] Ä o each For . ic h e ffxdpit of points fixed of set the Since . S  [X] soet-n n ae n ie on of point fixed any takes and one-to-one is i * respectively  ( , S 2.8. A M CU  nteHudrfmti sn as metric Hausdorff the in S 2 A ]b he iiecletoso o-mt com- non-empty of collections finite three be M] = )= 2 = , n hntetomp F maps two the then smxmlin maximal is * tflosthat follows It . . asayfxdpitof point fixed any maps f hnd then aeuiu ie onsA points fixed unique have X 9 [f tflosthat follows it , [ . [s 2 e [B Let lim . i 1 7 ( fteIS[X IFS the If f x . S i n 2 : )| H n . Ä U U }}} Ff H S ( 0 ehave we 0, sone-to-one. is fteIS[X IFS the If x  . X X ( X A  i # f Moreover A i 1 |1 i f wwÄ wwÄ n Ä n # C A] safntl eeal ustof subset generable finitely a is ) i 0 X 1 U i f S S , H | i = 2 X i  CU f ... 1 a nqefxdpitA point fixed unique a has X i S f let , n A i i [[x] [X]  ; 2 . ( D CU CU mle httefollowing the that implies steuiu ie point fixed unique the is * f X A f }}} fC If i , n 1 i oti es ie point fixed least this so , )< M] sa lmn of element an is ) o n A any for X ..., , X X i = f | n | ; nteohrhand, other the On . i x i , = # f 1 , Ä S f . D # 1 i N F 7 [C 2 ..., , F  A] i : swal hyper- weakly is ] } # }}} N n U : oafxdpoint fixed a to n ] i U 0 . *= 1 |1 f X f B = # N X F i H Ä sgvnby given is sweakly is ] U 7 F S and : Ä , X H X F N n U X , i . U n ] X n and X X ehave we 0 =  | Since . B X F Ä . CU S i the , M] F |< oa to H A n *, *. X X X = , POWER DOMAINS AND ITERATED FUNCTION SYSTEMS 189

n A*=n0 F X is indeed the unique fixed point of branch (fi1 fi2 }}}finX)n0 of the IFS tree at the node

F: HX Ä HX. Let A X be any non-empty compact fi1 fi2 }}}fim X which is then a leaf of the truncated tree as subset, and let =>0 be given. By Proposition 2.2.(iii), depicted in Fig. 3, and which contains the distinguished there exists m0 such that for all nm the diameters of point fi1 fi2 }}}fimx0#fi1 fi2 }}}fim X. n all the subsets in the collection f [X]=[fi1 fi2 }}}finX| By Proposition 2.2, the truncated tree will have finite n i1i2}}}in#7N] are less that =. Clearly, fi1 fi2 }}}finA depth. Let L= denote the set of all leaves of this finite fi1 fi2 }}}fin X and A* & fi1 fi2 }}}fin X fi1 fi2 }}}finX for all tree and let A= X be the set of all distinguished points of n n i1 i2 }}}in#7N. Therefore, by the lemma, dH(F A, A*)<=. the leaves. For each leaf l # L= , the attractor satisfies

K l$l & A*{< and A*=l # L= l&A*. On the other hand, for each leaf l # L= , we have l & A= {< and 2.3. Plotkin Power Domain Algorithm A==l # L= l & A= . It follows, by Lemma 2.8, that

Given a weakly hyperbolic IFS [X; f1, ..., fN], we want to dH(A= , A*)=. The algorithm therefore traverses the IFS formulate an algorithm to obtain a finite subset A= of X tree in some specific order to obtain the set of leaves L= and which approximates the attractor A* of the IFS up to a hence the finite set A= which is the required discrete given threshold =>0 with respect to the Hausdorff metric. approximation. We will make the assumption that for each node of the For a hyperbolic IFS and for X=A*, this algorithm IFS tree it is decidable whether or not the diameter of the reduces to that of Hepting et al. [24]. We will here obtain node is less than =. For a hyperbolic IFS, we have an upper bound for the complexity of the algorithm when the maps fi are contracting affine transformations as this is always the case in image compression. First, we note that | fi1 fi2 }}}fin X|si1 si2 }}}sin |X|, there is a simple formula for the contractivity of an affine 2 2 where s is the contractivity factor of f , and, therefore, the map. In fact, suppose the map f: R Ä R is given at the i i 2 above relation is clearly decidable. However, there are other point z # R in matrix notation by z [ Wz+t, where the 2 interesting cases in applications where this relation is also 2_2 matrix W is the linear part and t # R is the translation decidable. For example, if X=[0, 1]n Rn and if, for every part of f. Then, the infimum of numbers c with n i # 7N , each of the coordinates of the map fi :[0,1] Ä [0, 1]n is, say, monotonically increasing in each of its | f(z)&f(z$)|c |z&z$| arguments, then the diameter of any node is easily com- puted as is the greatest eigenvalue (in absolute value) of the matrix W tW, where W t is the transpose of W [12]. This greatest n | fi1 }}}fin[0, 1] |=d(fi1 }}}fin(0, ..., 0), fi1 }}}fin(1, ..., 1)), eigenvalue is easily calculated for the matrix where d is the Eucleadian distance. It is then clear that the a b above relation is decidable in this case. W= , c d Let =>0 be given and fix x # X. We construct a finite \ + 0 to be given by subtree of the IFS tree as follows. For any infinite sequence | i1 i2 }}} #7N, the sequence ( | fi1 fi2 }}}finX|)n0 is decreas- ing and tends to zero, and, therefore, there is a least integer - :+;+- (:&;)2+#2, m0 such that | fi1 fi2 }}}fim X|=. We truncate the infinite where :=(a2+c2)Â2, ;=(b2+d2)Â2, and #=ab+cd.Iffis contracting then this number is strictly less than one and is the contractivity of f. While traversing the tree,

the algorithm recursively computes fi1 fi2 }}}finx0 and

si1 si2 }}}sin |X|, and if si1 si2 }}}sin |X|=, then the point

fi1 fi2 }}}finx0 is taken to belong to A= . An upper bound for the height of the truncated tree is obtained as follows. We n have si1 si2 }}}sins , where s=max1iN si<1 is the con- tractivity of the IFS. Therefore the least integer h with sh |X |= is an upper bound, i.e., h=Wlog (=Â|X |)Âlog sX, where WaX is the least non-negative integer greater than or equal to a. A simple counting shows that there are at most nine arithmetic computations at each node. There- fore, the total number of computations is at most 9(N+N2+ FIG. 3. A branch of the truncated IFS tree. N 3+}}}+Nh)=9(Nh+1&1)Â(N&1), which is O(N h).

File: 643J 253408 . By:MC . Date:12:02:96 . Time:08:53 LOP8M. V8.0. Page 01:01 Codes: 6016 Signs: 4215 . Length: 56 pic 0 pts, 236 mm File: 643J 253409 . By:BV . Date:15:02:96 . Time:16:20 LOP8M. V8.0. Page 01:01 Codes: 5543 Signs: 4004 . Length: 56 pic 0 pts, 236 mm ie ntrsof terms in given sueta o ahpitin by point each given for is that of assume pixels array nearest two-dimensional between tance a into digitised screen, r 1] computer 1]_[0, the [0, that that resolution assume furthermore, with can Suppose, we 1]. [0, necessary, if rescaling rt approximation crete for if i.e., integers convergence, positive of all rate constructive uniform a is there [0, normalised the on 3.1. based defini- is basic hyper- the which recall an tions. first We [15] for domain. probabilities power in result probabilistic with IFS corresponding IFS the bolic hyperbolic generalizing weakly by a of measure lim of convergence of rate one the IFS, on hyperbolic weakly information a needs of attractor the of generation te ad vni h aeo yeblcIS h com- the IFS, as hyperbolic grows a algorithm of the case of the plexity threshold in discrete even the hand, other of value the whatever a for process, (i.e., digitization the in resolution error given the case, worst the In that see to that easy is it pixel, 190 ntesre.Let screen. the on ie h pia aac ewe cuayadcomplexity taking and by accuracy reached between is balance optimal the tice, sequences o hserr ups ehv ekyhproi IFS hyperbolic weakly a have we Suppose [X error. this for attractor the approximating in ekyhproi F in of IFS attractor hyperbolic the weakly to a approximation discrete the screen, puter IFSs. hyperbolic of case the to similar result plexity on n[,1_0 ]i tmost at is 1] 1]_[0, [0, in point | _ f i A epoeteeitneaduiuns fteinvariant the of uniqueness and existence the prove We the for analysis complexity similar a have to order In etw osdrtepolmo ltig ntecom- the on plotting, of problem the consider we Next 1 ; r f d f i rbblsi oe Domain Power Probabilistic $ ies ergr ahpxla on ota h dis- the that so point a as pixel each regard We pixels. 2 valuation 1 H }}} hc satisfies: which ) ..., , iie ytedaee ftesre 0 ][,1]) 1]_[0, [0, screen the of diameter the by divided ( A f = $ i f n .IVRATMAUEO NIFS AN OF MEASURE INVARIANT 3. , i N X 1 A ] i = o l sequences all for |=0 *) 2 with } # }}} ntplgclspace topological on r r _ IHPROBABILITIES WITH _ d m A H m r 7 X uhta | that such , r = $ = ( ftesre,i tleast at is screen, the of A srpeetdb h ntsquare unit the by represented is , hr sa integer an is there , | N A etesto iespotd ic any Since plotted. pixels of set the be ob fteodrof order the of be to = hnoecnesl banacom- a obtain easily can one then , = $ R nvtbypoue ute error further a produces inevitably , A 2 yatasaino rgnand origin of translation a By . d = R )+ H A ( 2 A h iiiaino h dis- the of digitization The . A = d = $ h ers ie splotted is pixel nearest the .W iloti bound a obtain will We *. f H , i N 1 ( A - A f i &log i 2 = 1 = Y ) 2 }}} i , 2 A $ = smap is } # }}} n f *) - rmisnearest its from 2 i = as $ n X $ =1 2 - n =1 = |1 $ 7 ( - = Ä 2 m X 0 nthe On >0. 2 N |  .I follows It 2. ( $ &  2 [,1]_ [0, ,explicitly ), nfc,if fact, In . .I prac- In 0. r ( : Âm r 1.We &1). 2= $ 0 &1). + ( BA EDALAT ABBAS o all for Y $ = n . ) Ä - Ä 2 esof sets ipevaluation simple tion htwhenever that element if Moreover, otmelement bottom n iielna combination linear finite Any ipersaig tsae h bv rprisof properties above the shares it rescaling, simple hr for where autos[7 6 6.Teeoe any Therefore, 16]. 26, [27, valuations fpitvaluations point of hc h u fadrce set directed a of lub the which oaBrlmaueon measure by extended nience Borel uniquely be a can to [34] Saheb-Djahromi of lemma an case tions space topological h coo autoswt oa mass total with valuations of dcpo the (1 [+ The h ata re ( order partial The o any For A o w ipevaluations simple two For (iii) (ii) |- # (i) $ i P c continuous  hi fnraie ipevlain n,hneb a by hence and, valuations simple normalised of chain & + b 0i,o ore trivial. course, of is, =0 Y Y : = on omlsdpoaiitcpwrdomain power probabilistic normalised C n | a & & 0 = then , + ,i otnosvlainon valuation continuous a is ), ( ( + O a < ( then , ( Y & b Y Y )+ swl 3,p 4.Fr0 For 24]. p. [34, well as b # f o l pnsets open all for iff # =,and )=0, with ) )= O + 0 Y Y Ä & 1 A . ( ( the , auto 3,2,2]i auto such valuation a is 26] 27, [31, valuation = $ & Y san is c] b P [0, $ ( b & Y & )= a ( = b 1 ehave we ) ( Since . : $ O 0 # Y \ ) Y ossso h e fcniuu valua- continuous of set the of consists B b & O n a ai ossigo simple of consisting basis a has and on valuation point )= ( i & ( . = n sodrda follows: as ordered is and )=1 sas an also is Y ihcntn coefficients constant with # P r O ( & |- eie by defined ) A a b 1 ( sadrce e wt)o open of (wrt ) set directed a is ) )=sup { Y b $ _ O i otnosdp ihabottom a with dcpo continuous : Y =1 1, 0, ). b n , P b + = C c i )+ sup = hc ednt o conve- for denote we which r # Y i I (+ sadp ihbto in bottom with dcpo a is ) $ O sotie from obtained is + + & O b |- if otherwise. # ( i i 2 i A ) a ( = otnosdp iha with dcpo continuous of O i & & ae at based # b c ( ). I : Y O # b # sgvnby given is c C ), + , O ). 1 let 1, + s # , c ( P O $ Y 1 c c ) ecl ta it call we ; b Y i.e., , , stevalua- the is stelbof lub the is P r & P i ( c P  #[0, Y O 1 P 1 Y i Y ). 1 denote P ,ofa Y + c i ya by the ; Y = = & ), , File: 643J 253410 . By:BV . Date:15:02:96 . Time:16:20 LOP8M. V8.0. Page 01:01 Codes: 5725 Signs: 3679 . Length: 56 pic 0 pts, 236 mm h set the rpsto .8;w eoetesto l autoswhich valuations all in of supported set are the denote we 5.18]; Proposition set sspotdin supported is ntecmatmti space metric compact the on an and igeo map singleton an total 3.2. of conservation as regarded be mass. can property above 2,16]: [27, M and in h number the If set n,hne oe ust[5 oolr .] valuation A 5.9]. Corollary [15, + subset Borel a hence, and, orewt mass with source eldfndadidc nioopimbetween isomorphism an induce and well-defined Let neighbourhood points o-eaienumber non-negative a # + Let o let Now Theorem For 1 P |- X P a |- M sspotdin supported is 1 t oe o F ihProbabilities with IFS for Model . Y 1 b otnosdp ihbottom with dcpo continuous U , otnosdp ihbottom with dcpo continuous 1 [X + where , c s X X X ( 0implies {0 y # X h set the , ; + fnraie oe esrson measures Borel normalised of # ssi obe to said is M f fmxmleeet of elements maximal of ) X 1 1 s t ..., , = ( 1 ..[5 hoe 5.21]. Theorem [15, 3.1. C eacmatmti pc ota ( that so space metric compact a be bc X X B + s c uhthat such ) stefo fms from mass of flow the as n noe subset open an and + ( : O , # : s f s a C ( N 2 C X ( r )= X s X ; f,frall for iff, b ( t Ä # a s p any , H samxmleeetof element maximal a is ) b b U )by ( , e P )= 1 e c X = C : U : supported j ..., , X ( f = : ( M P + t ,te the then ), X Y S b [[x] of )( 1 r S c , c 1 1 U c b ,w aeb the by have we ), ecncnie any consider can We . p 1 X with s X + & # X N y ( uhthat such X + a [ [ b Ä Ä n lmn of element Any . C ] ecnidentify can We . X ))= [ Ä # | ea F ihprobabilities with IFS an be Define . + sasn ihmass with sink a as S & M + x b s B in OE OAN N TRTDFNTO SYSTEMS FUNCTION ITERATED AND DOMAINS POWER ( P H : 1 b ( # b O # X x B 1 e s n all and 1 s ( X s support . U ( )= a] > o n Scott- any for )>0 &1 + a ( U t + X X b ) X , )( X Therefore, . c [x] )if $ h aseadjare j and e maps The g = o n pnsub- open any For . X X U , ealta the that Recall . a c + s b X c of .(6) ). ( # embeds to pitn lemma splitting U sa is C X + X P S hr exists there , c stestof set the is P 1 1 hn the Then, . " sfollows. as X U U s G 1 b ( U S P X X ihthe with X # $ X 1 s X 1 , and X B subset c ))=0. which U $ and , onto [15, X sa as )is (5) is fteISte aedaeessrcl esthan less strictly diameters all have tree IFS the of h aewya h akvoperator Markov the as way same the esta 1 than less ec,fralfnt sequences finite all for Hence, level on tree IFS the of and o intersect not sSotcniuu n a,teeoe es ie point fixed least a therefore, by has, given and Scott-continuous is imtr falndso h F reo level on tree IFS the of nodes all of diameters ete at centred have O k by olcino l pnballs open all of collection therefore, that oto & of port pnsbesof subsets open U Since i ie on & point fixed have: we Furthermore, rnho h F tree: IFS the of branch & hwthat show tflosthat follows It of h trco fteIFS the of attractor the ad if hand, integer 1 *( 1teeeit oeinteger some exists there 1 k X i Theorem Proof P 2 = H m s tflosthat follows it , } # }}} ( 1 [x] & X (  U f  *( & + i ( 1 *( n hrfr h nqefxdpito H of point fixed unique the therefore and X ))=inf )( i n ( k f s g oso that show To . # * x H H i ehv ( have we , ( g snti h upr of support the in not is 7 2 1 let 1, & I O X k sgvnby given is  *= }}} B Âk S m m N | & )= g B )1 o ahinteger each For ))=1. A k *( A * si h bv,let above, the in As . Then, . $ 3 x $ ( A k f ,teei noe ball open an is there *, b x stespotof support the is * . ( i X Then . X k  ofHisin s m  2 .Let *. x i 1 & ) ( = )( X Then . . ) *( X m ))=sup N i O =1 k B # & H ( ))=inf & 1 B i 1 o ekyhproi IFS hyperbolic weakly a For , *( k U [x] *( k O i H )= $ 2 ( m p m ..., , S ( x O n X k O H : i 0 x N m $ sanihorodbssof basis neighbourhood a is A hrfr,frall for Therefore, . + ) 0b uhta h oe nlevel on nodes the that such be 0 (O n . k i [x] ) sup ))= f k X si h upr of support the in is ( n $ m i S )=sup *= 1 .B rpsto .(i) o each for 2.2(iii), Proposition By ). aedaee tityls hn1 than less strictly diameter have and where , =1 f f ( X $ k 1 i 2 X & 1 H k X b i &1 )( X [x] }}} *# ) m i )( . [[x] p B m : eteoe alo ais1 radius of ball open the be f k stelbo oeinfinite some of lub the is 0 ( Hence & i i 1 g m O 1 $ s $ *( S = ( X p n i X ( m X ) oethat Note )). x 1 B 1 X ( # i 0sc htaltenodes the all that such 0 0 O 2 & X sadcesn euneof sequence decreasing a is )( B }}} & )0adi olw that follows it and ))=0 O  frdu esta 1 than less radius of k *. }}} | )= ,let *, ( ti ufcett show to sufficient is it , k g $ k x k n , x =,a eurd To required. as )=1, ( ( i n p 0 ti aia element maximal a is it H x 1 let 1, p )) m B # B i 0b uhta the that such be 0  1 K i ))=0, T m k A # p m $ f ( k x $ ( i i * x 1 nE.() Then, (3). Eq. in 7 2 1 $ x p f m }}} # hr A where ] i )) }}} X 1 ) m N i 1 f &  )( A i O 2 }}} .O h other the On *. with (b p f }}} H O n,frany for and, * k i X n i n m f n Therefore, . p , k X =1. i i sdfndin defined is $ m ) =,and, )=1, hc does which i r strictly are n X hn for Then, . , >0. i m . o some for # .(7) h least the h sup- The I *  k [x] Âk ethe be n Let . ,we Xis 191 Âk Âk H in n . 192 ABBAS EDALAT

FIG. 4. The IFS tree with transitional probabilities.

Corollary 3.3. For a weakly hyperbolic IFS, the nor- weighted leaves of the truncated IFS tree represents a simple malised measure +*=j(&*) # M1X is the unique fixed point of valuation which is a discrete approximation to the invariant 1 1 $ the Markov operator T: M X Ä M X. Its support is the measure +*. Then the total mass given to each pixel in A= is unique attractor A* of the IFS. the sum of the masses of all leaves corresponding to that pixel. 3.3. Probabilistic Power Domain Algorithm In the hyperbolic case, the probabilistic algorithm traver-

ses the finite tree and recursively computes fi1 fi2 }}}finx0, Since the Plotkin power domain algorithm in Section 2.3 pi1pi2 }}}pin and si1 si2 }}}sin |X|, and if si1 si2 }}}sin |X|=, provides a digitised discrete approximation A$ to the attrac- = then the weight of the pixel for fi1 fi2 }}}fin x0 is incremented tor A*, the question is how to render the pixels in A$ to = by pi1 pi2 }}}pin. A simple counting shows that this takes obtain an approximation to the invariant measure +*. We at most 10 arithmetic computations at each node. There- now describe an algorithm to do this, which extends that of fore, the total number of computations is at most Hepting et al. for a hyperbolic IFS with probabilities [24]. 10(N+N 2+N 3+}}}+Nh), which is O(N h) as before. Assume again that the unit square represents the digitised screen with r_r pixels. Suppose [X; f1, ..., fN ; p1 , ..., pN] is 4. A MODEL FOR RECURRENT IFS a weakly hyperbolic IFS with X [0, 1]_[0, 1] and =>0 is the discrete threshold. Fix x0 # X. The simple valuation In this section, we will construct a domain-theoretic m H $X of Eq. (7) can be depicted by the mth level of the IFS model for weakly hyperbolic recurrent IFSs. Assume that tree labelled with transitional probabilities as in Fig. 4. [X; f1, ..., fN] is an IFS and ( pij)(1i,jN)isan The root X of the tree has mass one and represents indecomposable row-stochastic matrix. Then [X; fj; pij ;

$X . Any edge going from a node t(X ), where t= fi1 b i, j=1, 2, ..., N] is a recurrent IFS. We will see below that fi2 b }}} bfim is a finite composition of the maps fi , to its child this gives rise to a Markov chain on the coproduct of N t( fi (X )) is labelled with transitional probability pi for copies of X. (See [20] for an introduction to Markov i=1, ..., N. The transitional probability label on each edge chains.) gives the flow of mass from the parent node (source) to the N For a topological space Y, we let Y = j=1 Y_[j] child node (sink) in the sense of Eq. (5) in the splitting denote the coproduct (disjoint sum) of N copies of Y [37], lemma. The total mass of the node fi1 fi2 }}}fimX on level m i.e., is, therefore, the product p p }}}p of the labels of all the i1 i2 im N edges leading from the root to the node, in agreement with Y = : Y_[j]=[(y,j)|y#Y,1jN], m the expansion of H $X in Eq. (7). We again make the j=1 assumption that it is decidable that the diameter of any node is less than = or not. The algorithm then proceeds, as with its frame of open sets given by 0(Y )=(0(Y))N, and its in the deterministic case, to find all the leaves of the IFS tree Borel subsets by B(Y )=(B(Y))N, where B(Y ) is the set of and, this time, computes the mass of each leaf. The set of all Borel subsets of Y.

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Markov fixed unique a generalised a for the indeed that has IFS operator generally recurrent more stationary hyperbolic here unique weakly show a will have We does distribution. IFS recurrent hyperbolic a hc a ewitnas written be can which  anlydfnstegnrlsdHthno metric Hutchinson generalised the defines Barnsley o ie ( given that a for given is proof no Nevertheless, ino h xetdvleo elvle ucin on functions real-valued of distribu- value in expected convergence the shown the of is proving tion it by hand, 2.1] other Theorem the On [4, verified. in not is point fixed a of as O o some for oethat Note otatvt fteISbtas ntemti ( matrix the on also the but on the IFS only not expects the depend notes one of will he However, factor that contractivity point. contractivity fixed 406] the unique and that a p. map contracting have a to [1, be therefore to in operator Markov states generalised then and hc sdfndbelow. defined is which of copies ping M  rbblt vector probability there- and chain, have: Markov we finite fore, ergodic an of matrix itional  =( N N j j ewl civ h bv ak ihu n edfra for need any without task, above the achieve will We oeta u supin ml ht( that imply assumptions our that Note If Proposition 1 =1 =1  X rÄ X  H h nqeFxdPito h akvOperator Markov the of Point Fixed Unique The Y N j =1 O T by m & ( where  + satplgclsae valuation a space, topological a is ( j j Ä ( j ) + , =1 p O Ä j =( &Ä ) U ij T )=sup j 1fr1 for =1 X c 2,p 0.W ilwr nasbcoof subdcpo a in work will We 90]. p. [26, )  X  j p = = swl-eie since well-defined is hc aife m satisfies which U O . ij (0 ;sbeunl,teeitneaduniqueness and existence the subsequently, ); 1 X i j : : , =1 =1 N N . 2,p 100]. p. [28, 4.1 = O { | + 2 c  i : =1 i f N ..., , : =1 1and 1 i N i &Ä ( N j ( ( =1 X m i x : p  \ 0 )& O ij )=1 j &Ä | ) ( + ( N X =( U N Y i  f o yeblcrcretIFS, recurrent hyperbolic a For . )#( f ( ihm with X i i ) f ( & d+ j Ä )_ y j &1 = j  ) 0 )| j i 0 1] [0, =( &  N j X ( [j] =1 Y N i )= | =1 d & X )) j T c ( 0(1 >0 hr xssaunique a exists There 1  x stecpoutof coproduct the is f j N ..., , 1,sc htfor that such =1), m i side contracting indeed is i , : =1 N d& y i ehave we p ,1 ), + & i j ij : =1 +} N N p . # with ) ij P stetrans- the is ) p f i 1 i ij j : Y   +  X i samap- a is N ( N & Ä X &Ä j = ) X ( # P ) rà R  O H  P 1 that , p U and 193 )= c on ij j N Y X ). File: 643J 253413 . By:BV . Date:15:02:96 . Time:16:20 LOP8M. V8.0. Page 01:01 Codes: 6061 Signs: 2696 . Length: 56 pic 0 pts, 236 mm h eeaie akvoeao on operator Markov generalised the q 8.If (8). Eq. ehv ( have we 194 n eietetomaps two the define and Let sÄ on measure Borel a on measure Borel bottom with > implies which have we & for that Note where rpsto ..Put 4.1. Proposition where ete aetefloiggnrlsto fTerm3.1. Theorem of generalisation following the have then We U of embedding the be iea smrhs between isomorphism an give j ( ( X B ie eurn IFS recurrent a Given Let Let Theorem N j U  =1 n oe subset Borel Any . )=( M X eÄ P & ( 1 0 )= H m H : 0 X  s m   M ( # ( j j ( H = &Ä B ( &Ä m  &Ä P (1 U )( 1= 0 1 )) ( j [ )) # j 1 &Ä X + O X 4 & Ä U  j ( o 1 for ))  &Ä P j ( S . & 0 [ Ä )= 2 of =( U .Therefore, ). j P X n n ( any and , & j j : 0 1 =1 0 1 j N ) = .  j U U j 0 1 X ( S + U egvnby given be U Ä # U  &  X X m 0 1 N b )= h w ase and eÄ maps two The X 1 M U X = P X sÄ N j then , = P H , N j i =1 j =1 steuiu rbblt etrin vector probability unique the is ) &1  X ic each since  = 0 1 X  & ( )= 1 0 1  i i x : U X 2 B : : [&Ä =1 =1 sÄ U N N   notesto aia lmnsof elements maximal of set the onto j P  , ..., , [&Ä N : : p =1 =( N X | [X j X m X 1 ij # & & since , N p i p )  U =1 H & . j # j ij P j ij [ & seach as Ä (  ) & P i B ; tas olw that follows also It . X and X j j ( & P & &Ä b N 0 1 M f ( &Ä & j 0 1 # i p i U i U f ) U [ ( 1 Ä )# )= ( b )# 0 U ; ij j U [x] U P j &1 =( s 0 1 X p X f X & of ( X & X P X H 0 1 X PU ij i B j &1 P and  m 0 (  )= U | ; si h eiiinof definition the in as 1 X san is m &Ä = ( , j | C 0 1 f )= U saBrlsbe of subset Borel a is )  i j &Ä ( & X ( j & U , ,1 j &1 } X U 1 ÄX sÄ ), à nue oe subset Borel a induces 0 ) P j X j &Ä : X 1 ..., =1, xed nqeyt a to uniquely extends = $ &Ä C (  X } S i 1 xed nqeyto uniquely extends ehave we r eldfndand well-defined are à : O =1 S U X )=1 N U since , iff |- 0 1 ) 0 1 , j U X j X Ä] &Ä ;i te words, other in ); U  m p m otnosdcpo continuous X )1, &Ä ij X by . 2 ] j N] m  [ Ä . $ N] X i &Ä = & n let and , &Ä M ..., , j # ( b eextend we m U P P 0 1 sÄ . X j 0 1 X m 0 1  . U BA EDALAT ABBAS U N and ) X X U T $  = X X iff in ) . oe( fore h eeaie akvoeao a es ie point. fixed least a has operator Markov generalised the eie sfollows: as defined sin is oeta yPooiin4.1, Proposition by that Note a o hwb nuto that induction by show now can hn o any for Then, continuous H rniinlpoaiiymti 2,p 1]( 414] p. [20, matrix probability transitional ro for proof reuil,adsatisfies and irreducible, h ct-otniyof Scott-continuity The yPooiin41 ehave we 4.1, Proposition By smntn.Let monotone. is &Ä  *= hn o each for Then, . Lemma IFS, recurrent any for that, shows lemma following The Proposition Proof Proof e sfn nepii oml o h es ie point fixed least the for formula explicit an find us Let P  0 1 q n \ U ti meitl enfo h eiiinthat definition the from seen immediately is It . Let . ij ( H H swl-eie;i saanrow-stochastic, again is it well-defined; is ) H X T    .  4 n ' ( n . ( ssimilar. is k 4 & respectively & . 0 0 &Ä O )of & ) &Ä  k j j =( 4 = =( + h apn H mapping The ( . U (&Ä 3 ( j _ . O 1 X & # H i O ,  k  1 i )= $ 2 [ ) )=( ti ovnett s h inverse the use to convenient is It . j , ..., , =sup = =sup = n ie on fH of point fixed Any ) = = f ,2 ..., 2, 1, & k j j : f K N 0 2 H i i q # 1 ' j j M  n  i i ..., , : : =1 =1 k N f N : : =1 =1 ij &1 H N N k k 0 i 2 = follows. &Ä N i ea nraigcanin chain increasing an be 0 1 }}} =1 ( =1 U X H i i ( p p j : ) : =1 =1 f & & m m  : N N =1 &Ä H N m X j i ij ij N] n j N ). &Ä ( m m &1 ( & j i & U ehave we ,  )# j k U p p p 0fr1 for >0 i i i i j )( : k X ( : =1 ( q N ij X ij ji q X ehave we U )( P (10) . f O ij P .(9) ji sup \  ) = O j &1 1 0 1 )= X p k ' 1 ).  q U k U K ij ) ). i m ( 1 & X & X U & i m 2 i k j k i i k }}} ( Ä X ( for j eafxdpitof point fixed a be + srqie.The required. as ,  f f )) j &1 ( P j j &1 q (  epcieyT respectively j f i 0 1 n 1 ..., =1, O &2 j &1 U O N q j i ij sScott- is X j ) n O ) &1 hc is which ) n there- and j ) P N 0 1 .We U X  H  ) . POWER DOMAINS AND ITERATED FUNCTION SYSTEMS 195

FIG. 5. The recurrent IFS tree with tansitional probabilities.

In fact, is unique. Using the explicit form of &* in Eq. (10), we can show as in the corresponding proof for a weakly hyperbolic N IFS with probabilities (Theorem 3.2) that &Ä *#S1UX.It &1 0 (H&0)j=:pij mi $X b f j 1 then follows that &Ä * is maximal in P0 UX, and hence is the i=1 unique fixed point. By Eq. (10), the support of &Ä * is indeed N (S(A* & fj X ))j. K = : pij mi $fjX i=1 It then follows, similar to the case of an IFS with =m $ . j fj X probabilities, that }Ã (&Ä *) is the unique stationary distribution &Ä * of the generalised Markov operator T : M1X Ä M1X of Assuming the result holds for n, we have Subsection 4.1, and that the support of +Ä *is(A*&fjX))j.

n+1 n (H &0)j=(H (H &0))j 4.3. The Recurrent Probabilistic Power Domain Algorithm N N Theorem 4.5 provides us with the recurrent algorithm to = : pij : mi qii1 }}}qin&2 in&1 i=1 i1, ..., in&1=1 generate the stationary distribution of a recurrent IFS on

&1 the digitised screen. Given the recurrent IFS [X; fj; pij ; _($f f }}}f X)b f j i i1 in&1 i, j=1, 2, ..., N], where X is contained in the unit square, N consider the recurrent IFS tree with transitional = : mi pij qii }}}qi i $f f f }}}f X 1 n&2 n&1 j i i1 in&1 probabilities in Fig. 5. Let =>0 be the discrete threshold. i, i1, ..., in&1 N Initially, the set X_[j]is given mass mj , which is then = : m q q }}}q $ , distributed amongst the nodes of the tree according to the j ji ii1 in&2 in&1 fj fi fi1 }}}fin&1 X i, i1, ..., in&1 inverse transitional probability matrix (qij). The algorithm first computes the unique stationary initial distribution as required. N (mi), by solving the equations mj=i=1 mi pij for mj Theorem 4.5. For a weakly hyperbolic recurrent IFS, (1jN) with the Gaussian elimination method, and the extended generalized Markov operator H : P1UX Ä determines the inverse transition probability matrix (qij) 1 X has a unique fixed point &Ä 1 X with support given by Eq. (9). The number of arithmetic computations P U *#S0U 3 (S(A* & f X )) where A* is the unique attractor of the IFS. for this is O(N ). Then the algorithm proceeds, exactly as j j the probabilistic algorithm, to compute, for each pixel, Proof. We know, by Proposition 4.3 that any fixed the sum of the weights mi1 qi1i2 }}}qin&1in of the leaves point of H is in P1 UX. Therefore, it is sufficient to show that 0 fi1 fi2 }}}finX of the IFS tree which occupy that pixel. The the least fixed point &Ä *of number of computations for the latter is O(N h) as before, where h=Wlog (=Â|X |)Âlog sX. Therefore, the complexity of 1 1 h$ H :P0UXÄP0UX the algorithm is O(N ) where h$=max(h,3).

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POWER DOMAINS AND ITERATED FUNCTION SYSTEMS 197

u 14. Dubuc, S., and Elqortobi, A. (1990), Approximations of fractal sets, Since S(g, &n) and gd+* both lie between S (g, &n) and Sl(g, & ), we conclude that J. Comput. Appl. Math. 29, 7989. n 15. Edalat, A. (1995), Dynamical systems, measures and fractals via domain theory, Inform. and Comput. 120, 3248. 16. Edalat, A. (1995), Domain theory and integration, Theor. Comput. Sci. S(g, & )& gd+Ä*=. n | 151, 163193. } } 17. Edalat, A. (1995), Domain theory in learning processes, in ``Proceed- ings of the Eleventh International Conference on Mathematical Foun- 1Âk Therefore, S(g, &n) with n=Wlog ((=Âc) Â|X |)Âlog sX is the dations of Programming Semantics'' (S. Brookes, M. Main, A. Melton, required approximation and the complexity is O(N n). and M. Mislove, Ed.), Electronic Notes in Theoretical Computer Science, Vol. 1, Elsevier, Amsterdam. 18. Elton, J.(1987), An ergodic theorem for iterated maps, J. Ergodic ACKNOWLEDGMENTS Theory Dynamical Systems 7, 481487. 19. Falconer, K. 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