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

Published online by Cambridge University Press:  15 September 2005

I. MELBOURNE  and
A. WINDSOR
I. MELBOURNE
Affiliation:
Department of Mathematics and Statistics, University of Surrey, Guildford GU2 7XH, UK (e-mail: ism@math.uh.edu)
A. WINDSOR
Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, TX 78712, USA (e-mail: awindsor@math.utexas.edu)
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Abstract

Let M be the four-dimensional compact manifold $M=T^2\times S^2$ and let $k\ge2$. We construct a $C^\infty$ diffeomorphism $F:M\to M$ with precisely k intermingled minimal attractors $A_1,\dotsc, A_k$. Moreover the union of the basins is a set of full Lebesgue measure. This means that Lebesgue almost every point in M lies in the basin of attraction of Aj for some j, but every non-empty open set in M has a positive measure intersection with each basin. We also construct $F:M\to M$ with a countable infinity of intermingled minimal attractors.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 25 , Issue 6 , December 2005 , pp. 1951 - 1959
Copyright
2005 Cambridge University Press

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