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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.

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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