Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-08T08:26:01.845Z Has data issue: false hasContentIssue false
  • English
  • Français

Orbit entropy in noninvertible mappings

Part of: Connections with other structures, applications Measure-theoretic ergodic theory

Published online by Cambridge University Press:  09 April 2009

Uhland Burkart
Uhland Burkart
Affiliation:
Fachbereich Mathematik, Universität Marburg, Lahnberge, D-3550 Marburg, Federal Republic of Germany
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Bassed on the intrinsic structure of a selfmapping T: S → S of an arbitrary set S, called the orbit-structure of T, a new entropy is defined. The idea is to use the number of preimages of an element x under the iterates of T to characterize the complexity of the transformation T and their orbit graphs. The fundamental properties of the orbit entropy related to iteration, iterative roots and iteration semigroups are studied. For continuous (differentiable) functions of Rn to Rn, the chaos of Li and Yorke is characterized by means of this entropy, mainly using the method of Straffingraphs.

MSC classification

Secondary: 54H20: Topological dynamics 28D20: Entropy and other invariants
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 40 , Issue 1 , February 1986 , pp. 95 - 110
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Adler, R. L., Konheim, A. G. and McAndrew, M. H., ‘Topological entropy’, Trans. Amer. Math. Soc. 114 (1965), 309–319.CrossRefGoogle Scholar
[2]Adler, R. L. and McAndrew, M. H., ‘The entropy of Chebyshev polynomials’, Trans. Amer. Math. Soc. 121 (1966), 236–241.CrossRefGoogle Scholar
[3]Beddington, J., Free, C. and Lawton, J., ‘Dynamic complexity in predator-prey models framed in difference equations’, Nature 255 (1975), 58–60.CrossRefGoogle Scholar
[4]Block, L., ‘Mappings of the interval with finitely many periodic points have zero entropy’, Proc. Amer. Math. Soc. 67 (1977), 357–361.Google Scholar
[5]Block, L., ‘Simple periodic orbits of mappings of the interval’, Trans. Amer. Math. Soc. 254 (1979), 391–398.Google Scholar
[6]Bowen, R. and Franks, J., ‘The periodic points of maps of the disk and the interval’, Topology 15 (1976), 337–342.CrossRefGoogle Scholar
[7]Burkart, U., Zur Charakterisierung diskreter dynamischer Systeme (Inaugural dissertation, Universität Marburg, 1981).Google Scholar
[8]Burkart, U.. ‘Interval mapping graphs and periodic points of continuous functions’, J. Combinatorial Theory Ser. B 32 (1982), 57–68.CrossRefGoogle Scholar
[9]Diamond, P., ‘Chaotic behaviour of systems of difference equations’, Internat. J. Systems Sci. 8 (1976), 953–956.Google Scholar
[10]Graw, R., Über die Orbitstruktur stetiger Abbildungen (Inaugural dissertation, Universität Marburg, 1981).Google Scholar
[11]Guckenheimer, J., Oster, G. F. and Ipaktchi, A., ‘The dynamics of density dependent population models’, J. Math. Biol. 4 (1977), 101–147.CrossRefGoogle ScholarPubMed
[12]Guckenheimer, J., ‘The bifurcation of quadratic functions’, Ann. New York Acad. Sci. 316 (1979), 78–85.CrossRefGoogle Scholar
[13]Hénon, M., ‘A two-dimensional mapping with a strange attractor’, Comm. Math. Phys. 50 (1976), 9–77.CrossRefGoogle Scholar
[14]Kloeden, P. E., ‘Chaotic difference equations are dense’, Bull. Austral. Math. Soc. 15 (1976), 371–379.CrossRefGoogle Scholar
[15]Kloeden, P. E., ‘Chaotic difference equations in Rn’, J. Ausiral. Math. Soc. (Ser. A) 31 (1981), 217–225.Google Scholar
[16]Kolmogorov, A. N. and Tihomirov, V. M., ‘ε-entropy and ε-capacity of sets in functional spaces’. Trans. Amer. Math. Soc. 17 (1961), 277–364.Google Scholar
[17]Li, T. Y. and Yorke, J. A., ‘Period three implies chaos’, Amer. Math. Monthly 82 (1975), 85–992.CrossRefGoogle Scholar
[18]Lorenz, E., ‘Deterministic nonperiodic flow’, J. Atmospheric Sci. 20 (1963), 130–141.2.0.CO;2>CrossRefGoogle Scholar
[19]Marotto, F. R., ‘Snap-back repellers imply chaos in Rn’, J. Math. Anal. Appl. 63 (1978), 199–223.CrossRefGoogle Scholar
[20]May, R. M., ‘Biological populations with nonoverlapping generations: stable points, stable cycles and chaos’, Science 186 (1974), 645–647.CrossRefGoogle ScholarPubMed
[21]May, R. M., ‘Bifurcations and dynamic complexity in ecological systems’, Ann. New York Acad. Sci. 316 (1979), 517–529.Google Scholar
[22]Mira, C., ‘Systemes a dynamique complexe et bifurcations de type “boites emboitées”, cas des récurrences d'ordre 1 determinées par une fonction a un seul extrémum’, Partie 1 et 2, R. A. I. R.O. Automat. 12 (1978), 63–94, 171–190.Google Scholar
[23]Misiurewicz, M. and Szlenk, W., ‘Entropy of piecewise monotone mappings’, Studia Math. 67 (1980), 45–63.CrossRefGoogle Scholar
[24]Myrberg, P. J., ‘Iteration der reellen Polynome zweiten Grades I’, Ann. Acad. Sci. Fenn. Ser. A I Math. 256 (1958), 1–10.Google Scholar
[25]Myrberg, P. J., ‘Iteration der reellen Polynome zweiten Grades II’, Ann. Acad. Sci. Fenn. Ser. A I Math. 268 (1959), 1–13.Google Scholar
[26]Nagashima, H., ‘Chaotic states in the Belousov-Zhabotinsky reaction’, J. Phys. Soc. Japan 49 (1980), 2427–2428.Google Scholar
[27]Poincaré, H., Les méthodes nouvelles de la méchanique céleste (Vol. 3, Gautbier-Villars, Paris, 1892).Google Scholar
[28]Rokhlin, V. A., ‘Lectures on the entropy theory of transformations with invariant measures’, Russian Math. Surveys 22 (1967), 1–52.Google Scholar
[29]Šarkovskii, A. N., ‘Coexistence of cycles of a continuous map of the line into itself’ (Russian), Ukrain. Mat. Ž. 16 (1964), 61–71.Google Scholar
[30]Straffin, Ph. D. Jr, ‘Periodic points of continuous functions’, Math. Mag. 51 (1978), 99–105.CrossRefGoogle Scholar
[31]Targonski, G., Topics in iteration theory (Studia Mathematica 6, Vandenhoeck and Ruprecht, Gottingen, Germany, 1981).Google Scholar