Intermingled Basins for the Triangle Map

James C. Alexander, Brian R. Hunt, Ittai Kan, and James A. Yorke

Ergod. Th. Dynam. Sys. 16 (1996), 651-662.

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.

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