Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T05:32:41.673Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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.

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

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

[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