Theory and Formal Methods 1993 pp 82-99 | Cite as

# Dynamical systems, Measures and Fractals via Domain Theory

## Abstract

We introduce domain theory in the computation of dynamical systems, iterated function systems (fractals) and measures. For a discrete dynamical system (*X, f*), given by the action of a continuous map *f*: *X* → *X* on a metric space *X*, we study the extended dynamical systems (*VX, Vf*) and (*UX, Uf*) where *V* is the Vietoris functor and *U* is the upper space functor. In fact, from the point of view of computing the attractors of (*X, f*), it is natural to study the other two systems: A compact attractor of (*X, f*) is a fixed point of (*VX, Vf*) and a fixed point of (*UX, Uf*). We show that if (*X, f*) is chaotic, then so is (*UX, Uf*). When *X* is locally compact *UX* is a continuous bounded complete dcpo. If *X* is second countable as well, then *UX* will be ω-continuous and can be given an effective structure. We show how strange attractors, attractors of iterated function systems (fractals) and Julia sets are obtained effectively as fixed points of deterministic functions on *UX* or fixed points of non-deterministic functions on *CUX* where *C* is the convex (Plotkin) power domain. We also establish an interesting link between measure theory and domain theory. We show that the set, *M*(*X*), of Borel measures on *X* can be embedded in *PUX*, where *P* is the probabilistic power domain. This provides an effective way of obtaining measures on *X*. We then prove that the invariant measure of an hyperbolic iterated function system with probabilities can be obtained as the unique fixed point of an associated continuous function on *PUX*.

### Keywords

Manifold Rubber Harness## Preview

Unable to display preview. Download preview PDF.

### References

- [1]J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey. On Devaney’s definition of chaos.
*The American mathematical monthly*, 99(4): 332–334, 1992.MathSciNetMATHCrossRefGoogle Scholar - [2]M. F. Barnsley.
*Fractals Everywhere*. Academic Press, 1988.Google Scholar - [3]M. F. Barnsley and S. Demko. Iterated function systems and the global construction of fractals.
*The proceedings of the royal society of London*, A399: 243–275, 1985.MathSciNetCrossRefGoogle Scholar - [4]M. F. Barnsley, A. Jacquin, L. Reuter, and A. D. Sloan. Harnessing chaos for image synthesis.
*Computer Graphics*,1988. SIGGRAPH proceedings.Google Scholar - [5]M. F. Barnsley and A. D. Sloan. A better way to compress images.
*Byte magazine*, pages 215–223, January 1988.Google Scholar - [6]P. Blanchard. Complex analytic dynamics on the Riemann sphere.
*Bull. A.M.S.*, 11: 85–141, 1984.MathSciNetMATHCrossRefGoogle Scholar - [7]L. Blum, M. Shub, and S. Smale. On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines.
*Bull. Amer. Math. Soc. (N.S.)*, 21: 1–46, 1989.MathSciNetMATHCrossRefGoogle Scholar - [8]P. C. Bressloff and J. Stark. Neural networks, learning automata and iterated function systems. In A. J. Crilly, R.. A. Earnshaw, and H. Jones, editors,
*Fractal and Chaos*, pages 145–164. Springer-Verlag, 1991.CrossRefGoogle Scholar - [9]R. Devaney.
*An Introduction to Chaotic Dynamical Systems*. Addison Wesley, second edition, 1989.MATHGoogle Scholar - [10]A. Edalat. Continuous information categories. Technical Report Doc 91/41, Department. of Computing, Imperial College, 1991.Google Scholar
- [11]A. Edalat and E. C. Zeeman. The stable classes and the codimension one bifurcations of the planar replicator system.
*Nonlinearity*, 5 (4): 921–939, 1992.MathSciNetMATHCrossRefGoogle Scholar - [12]M. J. Feigenbaum. The universal metric prperties of nonlinear transformation.
*J. Statistical Physics*,21:669–709, 1979MathSciNetCrossRefGoogle Scholar - [13]J. E. Hutchinson. Fractals and self-similarity.
*Indiana Univ. Math. J.*, 30: 713–747, 1981.MathSciNetMATHCrossRefGoogle Scholar - [14]C. Jones.
*Probabilistic Non-determinism*. PhD thesis, University of Edinburgh, 1989.Google Scholar - [15]S. Lang.
*Real Analysis*. Addison-Wesley, 1969.Google Scholar - [16]C. Moore. Generalized shifts: unpredictability and undecidability in dynamical systems.
*Nonlinearity*, 4: 199–230, 1991.MathSciNetMATHCrossRefGoogle Scholar - [17]G. D. Plotkin. Post-graduate lecture notes in advanced domain theory (incorporating the “Pisa Notes”). Dept. of Computer Science, Univ. of Edinburgh, 1981.Google Scholar
- [18]W. Rudin.
*Real and Complex Analysis*. McGraw-Hill, 1966.Google Scholar - [19]N. Saheb-Djahromi. Cpo’s of measures for non-determinism.
*Theoretical computing science*,12(1):19–37, 1980.MathSciNetCrossRefGoogle Scholar - [20]M. B. Smyth. Powerdomains and predicate transformers: a topological view. In J. Diaz, editor,
*Automata*,*Languages and Programming*, pages 662–675, Berlin, 1983. Springer-Verlag. Lecture Notes in Computer Science Vol. 154.CrossRefGoogle Scholar - [21]M. B. Smyth. Topology. In S. Abramsky, D. Gabbay, and T. Maibaum, editors,
*Handbook of Logic in Computer Science*, chapter 5. Oxford University Press, 1992.Google Scholar - [22]W. Szlenk.
*An Introduction to the Theory of Smooth Dynamical Systems*. John Wiley & Sons, 1984.Google Scholar - [23]K. R. Wicks.
*Fractals and Hyperspaces*, volume 1492 of Lecture noten in mathematics. Springer-Verlag, 1991.Google Scholar