WIAS Preprint No. 132, (1994)

How 1-dimensional hyperbolic attractors determine their basins


  • Bothe, Hans Günter

2010 Mathematics Subject Classification

  • 58F12 58F15


  • Hyperbolic attractors, basins




Two attractors Λi (i=1,2) of diffeomorphisms ƒi : Mi → Mi will be called intrinsically equivalent if there is a homeomorphism h: Λ1 → Λ2 satisfying ƒ2h = hƒ1. If we can find a homeomorphism g: WsΛ1 → WsΛ2 of the basins WsΛi of Λi such that ƒ2g = gƒ1, then we say that Λ1, Λ2 are basin equivalent. Let Λ1, Λ2 be transversely tame 1-dimensional hyperbolic attractors which are intrinsically equivalent. Then, if WsΛ1, WsΛ2 are orientable and m = dim M1 = dim M2 ≥ 4, it is shown that Λ1, Λ2 are basin equivalent, provided these attractors are regarded, for some positive integer k, as attractors of ƒk1, ƒk2 instead of ƒ1, ƒ2, respectively. This conclusion implies that WsΛ1, WsΛ2 are homeomorphic under a homeomorphism which maps Λ1 to Λ2 and the stable foliation 𝔚sΛ1 of WsΛ1 to the stable foliation 𝔚sΛ2 of WsΛ2. (To be transversely tame is a weak restriction; hence, roughly, speaking, these facts hold for "almost all" 1-dimensional hyperbolic attractors.) If transverse tameness and m ≥ 4 is dropped from the assumption, then still the cartesian products WsΛi x ℝ are homeomorphic with a homeomorphism which maps Λ1 x {0} to Λ2 x {0} and 𝔚sΛ1 x ℝ to 𝔚sΛ2 x ℝ.

Appeared in

  • Nonlinearity 9 (1996), No. 5, pp. 1173--1190

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