Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-26T10:25:28.575Z Has data issue: false hasContentIssue false
  • English
  • Français

Topological conjugation of Lorenz maps by β-transformations

Published online by Cambridge University Press:  24 October 2008

Paul Glendinning
Paul Glendinning
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge CB3 9EW
Get access Rights & Permissions [Opens in a new window]

Abstract

Necessary and sufficient conditions for a Lorenz map to be topologically conjugate to a piecewise linear map with constant slope (a β-transformation) are given, first in terms of kneading invariants of the maps and then in terms of the topological entropy restricted to basic sets. The dynamics of β-transformations is also described.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 107 , Issue 2 , March 1990 , pp. 401 - 413
Copyright
Copyright © Cambridge Philosophical Society 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Afraimovich, V. S., Bykov, V. V. and Shil'nikov, L. P.. On structurally unstable attracting limit sets of Lorenz attractor type. Trans. Mosc. Math. Soc. 44 (1983), 153–216.Google Scholar
[2]Gambaudo, J. M., Lanford, O. E. III and Tresser, C.. Dynamique symbolique des rotations. C.R. Acad. Sci. Paris Sér I Math. 299 (1984), 823–826.Google Scholar
[3]Glendinning, P.. The structure of β-transformations and conjugates to Lorenz maps. (Unpublished manuscript.)Google Scholar
[4]Glendinning, P. and Sparrow, C. T.. Prime and renormalisable kneading invariants of Lorenz maps. (Preprint, 1989.)Google Scholar
[5]Guckenheimer, J.. A strange, strange attractor. In The Hopf Bifurcation and its Applications (ed. Marsden, J. and McCracken, M.) Appl. Math. Sci. vol. 19 (Springer-Verlag, 1976).CrossRefGoogle Scholar
[6]Guckenheimer, J. and Holmes, P.. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Appl. Math. Sci. vol. 42 (Springer-Verlag, 1983).CrossRefGoogle Scholar
[7]Guckenheimer, J. and Williams, R. F.. Structural stability of Lorenz attractors. Inst. Hautes Études Sci. Publ. Math. 50 (1979), 307–320.CrossRefGoogle Scholar
[8]Hardy, G. H. and Wright, E. M.. An Introduction to the Theory of Numbers (Clarendon Press, 1965).Google Scholar
[9]Hofbauer, F.. The maximal measure for linear mod one transformations. J. London Math. Soc. 23 (1981), 92–112.CrossRefGoogle Scholar
[10]Hubbard, J. and Sparrow, C. T.. The classification of expanding interval maps with a single discontinuity. (Preprint, 1989.)Google Scholar
[11]Lorenz, E. N.. Deterministic non-periodic flow. J. Atmos. Sci. 20 (1963), 130–141.2.0.CO;2>CrossRefGoogle Scholar
[12]Milnor, J. and Thurston, W.. On iterated maps of the interval, I, II. (Unpublished notes, Princeton 1977).Google Scholar
[13]Palmer, M. R.. On the classification of measure preserving transformations of Lebesgue spaces. Ph.D. thesis, University of Warwick (1979).Google Scholar
[14]Parry, W.. Symbolic dynamics and transformations of the unit interval. Trans. Amer Math. Soc. 122 (1966), 368–378.CrossRefGoogle Scholar
[15]Parry, W.. The Lorenz attractor and a related population model. In Ergodic Theory (eds. Denker, M. and Jacobs, K.) Lecture Notes in Math. vol. 729 (Springer-Verlag, 1979).Google Scholar
[16]Rand, D.. The topological classification of Lorenz attractors. Math. Proc. Cambridge Philos. Soc. 83 (1978), 451–460.CrossRefGoogle Scholar
[17]Sparrow, C. T.. The Lorenz Equations: Bifurcations, Chaos and Strange Attractors. Appl. Math. Sci. vol. 41 (Springer-Verlag, 1982).CrossRefGoogle Scholar
[18]Swinnerton-Dyer, H. P. F.. The basic Lorenz list and Sparrow's conjecture A. J. London Math. Soc. 29 (1984), 509–520.CrossRefGoogle Scholar
[19]Veerman, P.. Symbolic dynamics and rotation numbers. Physica A 134 (1986), 543–576.CrossRefGoogle Scholar
[20]Williams, R. F.. The structure of Lorenz attractors. Inst. Hautes Études Sci. Publ. Math. 50 (1979), 321–347.CrossRefGoogle Scholar