Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-17T18:10:34.742Z Has data issue: false hasContentIssue false
  • English
  • Français

Self-generation of self-replicating maps of an interval

Published online by Cambridge University Press:  19 September 2008

William Parry
William Parry*
Affiliation:
From the Mathematics Institute, University of Warwick, England
*
†Address for correspondence: Professor William Parry, Mathematics Institute, University of Warwick, Coventry CV4 7AL, England.
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.

We construct C1 symmetric maps of [−1, 1] to itself, satisfying a condition similar to Feigenbaum's: -φ(x) = φ(-x), φ2(bx) = bφ(x), 0<b <1. Under certain conditions, the non-wandering set consists of the one-sided Morse minimal set together with 2n points of period 2n for each n = 1, 2, … The main significance of the construction is its simplicity. Given a certain piece of the map φ, the rest is generated by the required equation.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 1 , Issue 2 , June 1981 , pp. 197 - 208
Copyright
Copyright © Cambridge University Press 1981

References

REFERENCES

[1]Collet, P., Eckmann, J. P. & Kock, H.. Period doubling bifurcations for families of maps on ℝn. (To appear.)Google Scholar
[2]Feigenbaum, M. J.. Quantitative universality for a class of non-linear transformations. J. Stat. Physics 19 (1978), 25–52.CrossRefGoogle Scholar
[3]Guckenheimer, J.. A strange, strange attractor. In The Hopf Bifurcation (ed. Marsden, & McCracken, ) Appl. Math. Sci. Springer-Verlag: Berlin, 1976.Google Scholar
[4]Hofbauer, F.. The maximal measure for the transformation T:x→βx + α mod 1. Journal L.M.S. (in the press).Google Scholar
[5]Jonker, L. and Rand, D.. Bifurcations in one dimension I: The non-wandering set. Invent, math. 62 (1981), 347–365.CrossRefGoogle Scholar
[6]Kakutani, S.. Ergodic theory of shift transformations. Proc. Berkeley Symp. Math. Stat. 5. Prob. 11 (1967), 405–414.Google Scholar
[7]Keane, M.. Generalised Morse sequence. Z. Wahrscheinlichkeitstheorie verw. Geb. 10 (1968), 335–353.Google Scholar
[8]Milnor, J. & Thurston, W.. On iterated maps of the interval I and II. Princeton University and the Institute for Advanced Studies, Princeton preprints.Google Scholar
[9]Misiurewicz, M.. Structure of mappings of an interval with entropy. (To appear.)Google Scholar
[10]Palmer, M. R.. Mixing properties of linear mod 1 maps of the unit interval. (To appear.)Google Scholar
[11]Parry, W.. The Lorenz attractor and a related population model. Proceedings of Ergodic Theory conference. Oberwolfach 1978. Lecture Notes in Math. no. 729. Springer-Verlag: Berlin, 1978.Google Scholar
[12]Williams, R. F.. The structure of Lorenz attractors. Publ. Math. I.H.E.S. 50 (1979) 321–347.CrossRefGoogle Scholar