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Intermingled basins for the triangle map

Published online by Cambridge University Press:  19 September 2008

James C. Alexander ,
Brian R. Hunt ,
Ittai Kan  and
James A. Yorke
James C. Alexander
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA
Brian R. Hunt
Affiliation:
Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA
Ittai Kan
Affiliation:
Department of Mathematics, George Mason University, Fairfax, VA 22030, USA
James A. Yorke
Affiliation:
Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA
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Abstract

A family of quadratic maps of the plane has been found numerically for certain parameter values to have three attractors, in a triangular pattern, with ‘intermingled’ basins. This means that for every open set S, if the basin of attraction of one of the attractors intersects S in a set of positive Lebesgue measure, then so do the other two basins. In this paper we mathematically verify this observation for a particular parameter, and prove that our results hold for a set of parameters with positive Lebesgue measure.

Information

Type
Survey Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 4 , August 1996 , pp. 651 - 662
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

[AKYY]Alexander, J. C., Kan, I., Yorke, J. A. and You, Z.. Riddled Basins. Int. J. Bif. Chaos 2 (1992), 795–813.CrossRefGoogle Scholar
[B]Birkhoff, G. D.. Proof of the ergodic theorem. Proc. Nat. Acad. Sci. 17 (1931), 656–660.CrossRefGoogle ScholarPubMed
[CE]Collet, P. and Eckmann, J-P.. Iterated Maps on the Interval as Dynamical Systems. Birkhauser, Boston, 1980.Google Scholar
[H]Hunt, B. R.. Estimating invariant measures and Lyapunov exponents. Ergod. Th. & Dynam. Sys. pp. 735–749 of this issue.Google Scholar
[HSY]Hunt, B. R., Sauer, T. and Yorke, J. A.. Prevalence: a translation-invariant ‘almost every’ on infinite-dimensional spaces. Bull. Amer. Math. Soc. 27 (1992), 217–238;CrossRefGoogle Scholar
Prevalence: an addendum. Bull. Amer. Math. Soc. 28 (1993), 306–307.Google Scholar
[J]Jakobson, M. V.. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm. Math. Phys. 81 (1981), pp. 39–88.CrossRefGoogle Scholar
[K1]Kan, I.Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin. Bull. Amer. Math. Soc. 31 (1994), 68–74.CrossRefGoogle Scholar
[K2]Kan, I.Intermingled basins. Ergod. Th. & Dynam. Sys. (to appear).Google Scholar
[L]Lopes, A.Dynamics of real polynomials on the plane and triple point phase transition. Math. Comp. Mod. 13 (1990), 17–32.CrossRefGoogle Scholar
[O]Oseledec, V. I.. A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19 (1968), 197–231.Google Scholar
[PS]Pugh, C. and Shub, M.. Ergodic Attractors. Trans. Amer. Math. Soc. 312 (1989), 1–54.CrossRefGoogle Scholar
[R]Ruelle, D.Applications conservant une mesure absolument continue par rapport à dx sur [0, 1]. Comm. Math. Phys. 55 (1977), 47–51.CrossRefGoogle Scholar
[RS]Rychlik, M. and Sorets, E.. Regularity and other properties of absolutely continuous invariant measures for the quadratic family. Comm. Math. Phys. 150 (1992), 217–236.CrossRefGoogle Scholar