Power Domains and Iterated Function Systems

We introduce the notion of weakly hyperbolic iterated function system (IFS) on a compact metric space, which generalises that of hyperbolic IFS. Based on a domain-theoretic model, which uses the Plotkin power domain and the probabilistic power domain respectively, we prove the existence and uniqueness of the attractor of a weakly hyperbolic IFS and the invariant measure of a weakly hyperbolic IFS with probabilities, extending the classic results of Hutchinson for hyperbolic IFSs in this more general setting. We also present finite algorithms to obtain discrete and digitised approximations to the attractor and the invariant measure, extending the corresponding algorithms for hyperbolic IFSs. We then prove the existence and uniqueness of the invariant distribution of a weakly hyperbolic 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 expected value of almost everywhere continuous functions with respect to this distribution. For hyperbolic recurrent IFSs and Lipschitz maps, one can estimate the integral up to any threshold of accuracy.

[1]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[2]  J. Douglas Faires,et al.  Numerical Analysis , 1981 .

[3]  S. Vickers.  Topology via Logic , 1989 .

[4]  Michael F. Barnsley,et al.  A better way to compress images , 1988 .

[5]  Arnaud Jacquin,et al.  Harnessing chaos for image synthesis , 1988, SIGGRAPH.

[6]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[7]  M. Barnsley,et al.  Iterated function systems and the global construction of fractals , 1985, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[8]  Richard L. Burden,et al.  Numerical analysis: 4th ed , 1988 .

[9]  Abbas Edalat.  Domain theory in learning processes , 1995, MFPS.

[10]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[11]  Michael F. Barnsley,et al.  Fractals everywhere, 2nd Edition , 1993 .

[12]  Deterministic rendering of self-affine fractals , 1990 .

[13]  Yuval Fisher.  Fractal Image Compression , 1994 .

[14]  Susumu Hayashi,et al.  Self-similar Sets as Tarski's Fixed Points , 1985 .

[15]  Viggo Stoltenberg-Hansen,et al.  Mathematical theory of domains , 1994, Cambridge tracts in theoretical computer science.

[16]  S. Dubuc,et al.  Approximations of fractal sets , 1990 .

[17]  Abbas Edalat.  Domain Theory and Integration , 1995, Theor. Comput. Sci..

[18]  M. Barnsley,et al.  Recurrent iterated function systems , 1989 .

[19]  John G. Kemeny,et al.  Finite Markov Chains. , 1960 .

[20]  Dietmar Saupe,et al.  Rendering Methods for Iterated Function Systems , 1991 .

[21]  S. Kusuoka,et al.  Dirichlet forms on fractals and products of random matrices , 1989 .

[22]  Lyman P. Hurd,et al.  Fractal image compression , 1993 .

[23]  Paul C. Bressloff,et al.  Neural networks, learning automata and iterated function systems , 1991 .

[24]  Ulrich Behn,et al.  One-dimensional Markovian-field Ising model: physical properties and characteristics of the discrete stochastic mapping , 1988 .

[25]  Abbas Edalat,et al.  Dynamical Systems, Measures and Fractals via Domain Theory , 1993, Inf. Comput..

[26]  Michael F. Barnsley,et al.  Fractals everywhere , 1988 .

[27]  M. Barnsley,et al.  Solution of an inverse problem for fractals and other sets. , 1986, Proceedings of the National Academy of Sciences of the United States of America.

[28]  M. Barnsley,et al.  A new class of markov processes for image encoding , 1988, Advances in Applied Probability.

[29]  J. L. van Hemmen,et al.  Multifractality in forgetful memories , 1993 .

[30]  J. Miller.  Numerical Analysis , 1966, Nature.

[31]  C. Sparrow.  The Fractal Geometry of Nature , 1984 .

[32]  C. Jones,et al.  A probabilistic powerdomain of evaluations , 1989, [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science.

[33]  S. Krantz.  Fractal geometry , 1989 .

[34]  J. Elton.  An ergodic theorem for iterated maps , 1987, Ergodic Theory and Dynamical Systems.

[35]  Claire Jones,et al.  Probabilistic non-determinism , 1990 .

[36]  John G. Kemeny,et al.  Finite Markov chains , 1960 .

[37]  N. Saheb-Djahromi,et al.  CPO'S of Measures for Nondeterminism , 1980, Theor. Comput. Sci..