Power Domains and Iterated Function Systems Abbas Edalat Department of Computing Imp erial College of Science Technology and Medicine Queens Gate London SW BZ UK Abstract We intro duce the notion of weakly hyp erb olic iterated function system IFS on a compact metric space which generalises that of hyp erb olic IFS Based on a domaintheoretic mo del which uses the Plotkin p ower domain and the probabili sti c p ower domain resp ectively we prove the existence and uniqueness of the attractor of a weakly hyp erb olic IFS and the invariant measure of a weakly hyp erb olic IFS with probabili ties extending the classic results of Hutchinson for hyp erb olic IFSs in this more general setting We also present nite algorithms to obtain discrete and digitised approximation s to the attractor and the invariant measure extending the corresp onding algorithms for hyp erb olic IFSs We then prove the existence and uniqueness of the invariant distributio n of a weakly hyp erb olic recurrent IFS and obtain an algorithm to generate the invariant distribution on the digitised screen The generalised Riemann integral is used to provide a formula for the exp ected value of almost everywhere continuous functions with resp ect to this distribution For hyp erb olic recurrent IFSs and Lipschitz maps one can estimate the integral up to any threshold of accuracy Intro duction The theory of iterated function systems has b een an active area of research since the seminal work of Mandelbrot on fractals and selfsimilarity in nature in late seventies and early eighties The theory has found applications in diverse areas such as computer graphics image compression learning automata neural nets and statistical physics In this pap er we will b e mainly concerned with the basic theoretical work of Hutchinson and a numb er of algorithms based on this work We start by briey reviewing the classical work See for a comprehensive intro duction to iterated function systems and fractals Iterated Function Systems An iterated function system IFS fX f f f g on a top ological space X N is given by a nite set of continuous maps f X X i N If X is i Submitted to Information and Computation a complete metric space and the maps f are all contracting then the IFS is i said to b e hyperbolic For a complete metric space X let HX b e the complete metric space of all nonempty compact subsets of X with the Hausdor metric d dened by H d A B inf f j B A and A B g H where for a nonempty compact subset C X and the set C fx X j y C dx y g is the paral lel body of C An hyp erb olic IFS induces a map F HX HX dened by F A f A f A f A In fact F is also contracting N with contractivity factor s max s where s is the contractivity factor of i i i f i N The numb er s is called the contractivity of the IFS By the i contracting mapping theorem F has a unique xed p oint A in HX which is called the attractor of the IFS and we have n A lim F A n!1 for any nonempty compact subset A X The attractor is also called a selfsimilar set For applications in graphics and image compression it is assumed that X is the plane R and that the maps are contracting ane transformations Then the attractor is usually a fractal ie it has ne complicated and nonsmo oth lo cal structure some form of selfsimilarity and usually a nonintegral Hausdor dimension A nite algorithm to generate a discrete approximation to the attractor was rst obtained by Hepting et al in See also It is describ ed in Section IFS with Probabilities There is also a probabilistic version of the theory that pro duces invariant probability distributions and as a result coloured images in computer graphics An hyp erb olic IFS with probabilities fX f f p p g is an hyp erb olic N N IFS fX f f f g with X a compact metric space such that each f N i i N is assigned a probability p with i N X p p and i i i Then the Markov operator is dened by T M X M X on the set M X of normalised Borel measures on X It takes a Borel measure M X to a Borel measure T M X given by N X B p f T B i i i for any Borel subset B X When X is compact the Hutchinson metric r H can b e dened on M X as follows Z Z r sup f f d f d j f X R jf x f y j dx y x y X g H X X Then using some Banach space theory including Alaoglus theorem it is shown that the weak top ology and the Hutchinson metric top ology on M X coincide thereby making M X r a compact metric space If the IFS is H hyp erb olic T will b e a contracting map The unique xed p oint of T then denes a probability distribution on X whose supp ort is the attractor of fX f f g The measure is also called a selfsimilar measure or N n a multifractal When X R this invariant distribution gives dierent p oint densities in dierent regions of the attractor and using a colouring scheme one can colour the attractor accordingly A nite algorithm to generate a discrete approximation to this invariant measure and a formula for the value of the integral of a continuous function with resp ect to this measure were also obtained in they are describ ed in Sections and resp ectively The random iteration algorithm for an IFS with probabilities is based on the following ergo dic theorem of Elton Let fX f f p p g N N b e an IFS with probabilities on the compact metric space X and let x X b e any initial p oint Put f N g with the discrete top ology Cho ose i at random such that i is chosen with probability p Let x f x i i 1 Rep eat to obtain i and x f x f f x In this way construct the i i i 2 2 1 sequence hx i Supp ose B is a Borel subset of X such that B n n where is the invariant measure of the IFS and B is the b oundary of B Let Ln B b e the numb er of p oints in the set fx x x g B Then n Eltons Theorem says that with probability one ie for almost all sequences hx i we have n no Ln B B lim n!1 n Moreover for all continuous functions g X R we have the following convergence with probability one P Z n g x i i g d lim n!1 n which gives the exp ected value of g Recurrent IFS Recurrent iterated function systems generalise IFSs with probabilities as follows Let X b e a compact metric space and fX f f f g an N hyp erb olic IFS Let p b e an indecomp osable N N rowsto chastic matrix ij ie P N p for all i ij j p for all i j and ij for all i j there exist i i i with i i and i j such that n n p p p i i i i i i 1 2 2 3 n1 n Then fX f p i j N g is called an hyp erb olic recurrent IFS For j ij an hyp erb olic recurrent IFS consider a random walk on X as follows Sp ecify a starting p oint x X and a starting co de i Pick a numb er i such that p is the conditional probability that j is chosen and dene i j 0 x Then pick i such that p is the conditional probability x f i j i 1 1 x Continue to obtain the sequence f x f that j and put x f i i i 1 2 2 hx i The distribution of this sequence converges with probability one to n n a measure on X called the stationary distribution of the hyp erb olic recurrent IFS This generalises the theory of hyp erb olic IFSs with probabilities In fact if p p is indep endent of i then we obtain an hyp erb olic IFS with ij j probabilities the stationary distribution is then just the invariant measure and the random walk ab ove reduces to the random iteration algorithm The rst practical software system for fractal image compression Barnsleys VRIFS Vector Recurrent Iterated Function System which is an interactive image mo delling system is based on hyp erb olic recurrent IFSs Weakly hyp erb olic IFS In p ower domains were used to construct domaintheoretic mo dels for IFSs and IFSs with probabilities It was shown that the attractor of an hyp erb olic IFS on a compact metric space is obtained as the unique xed p oint of a continuous function on the Plotkin p ower domain of the upp er space Similarly the invariant measure of an hyp erb olic IFS with probabilities on a compact metric space is the xed p oint of a continuous function on the probabilistic p ower domain of the upp er space We will here intro duce the notion of a weakly hyp erb olic IFS Our denition is motivated by a numb er of applications for example in neural nets where one encounters IFSs which are not hyp erb olic This situation can arise for example in a compact interval X R if the IFS contains a smo oth map 0 0 f X X satisfying jf xj but not jf xj Let X d b e a compact metric space we denote the diameter of any set a X by jaj supfdx y j x y ag As b efore let f N g with the discrete top ology and let b e the set of all innite sequences i i i i for n with the pro duct top ology n Denition An IFS fX f f f g is weakly hyperbolic if for all innite N X j f f sequences i i we have lim jf i i n!1 i n 2 1 Weakly hyp erb olic IFSs generalise hyp erb olic IFSs since clearly a hyp erb olic IFS is weakly hyp erb olic One similarly denes a weakly hyperbolic IFS with probabilities and a weakly hyperbolic recurrent IFS Example The IFS f f f g with f f dened by x x f x 0 0 and f x f x is not hyp erb olic since f f but can b e shown to b e weakly hyp erb olic Since for a weakly hyp erb olic IFS the map F HX HX is not necessarily contracting one needs a dierent approach to prove the existence and uniqueness of the attractor in this more general setting In this pap er we