Intermingled basins for the triangle map

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.

A $C^{\infty}$ diffeomorphism with infinitely many intermingled basins

“…A proof is presented in [1]. They did not verify that the basins occupy a set of full measure, but did show that the regular parts of the basin (those characterized by typical Lyapunov exponents) are intermingled for a set of parameters with positive measure.…”

“…. , ϕ n } and for every x ∈ T 2 , there exists ν x n−1 in the simplex generated by the measures µ (1) n−1 , . .…”

“…We choose the cores C i,j large enough so that every vertical line in T 2 intersects every row of cores and, moreover, there exists a column i 0 such that the vertical line intersects C i 0 ,j for each j ≥ 1. For every x, the orbit {h where ν x is in the simplex of measures µ (1) , . .…”

Estimating invariant measures and Lyapunov exponents

“…Finally, in Section 4 we apply this procedure to a specific example: the quadratic map with a particular parameter value studied previously by Ruelle [15]. In particular, we estimate a time average for this map which can be shown to be a Lyapunov exponent for a two-dimensional map studied in [2,1]. In the latter paper it is vital to know rigorously that the Lyapunov exponent is negative.…”

“…One exponent will simply be the Lyapunov exponent of the one-dimensional map, and the other will reflect the rate of contraction or expansion transverse to A for the two-dimensional map. This planar map, originally studied in [12], has been of great interest recently as a fundamental example of the phenomenon of "riddled" and "intermingled" basins of attraction [2,1]. In order to verify mathematically the properties of this map which were discovered with a computer, it is necessary to verify that the "transverse" Lyapunov exponent for A is negative, which in particular implies that A attracts a set of positive Lebesgue dimensional measure.…”