×

Representations of the Whitehead manifold \(Wh^ 3\) and Julia sets. (English) Zbl 0878.57015

Summary: When one extends to the Whitehead manifold the collapsible pseudo-spine representation theorem for homotopy 3-spheres, the tight transversals to the set of double lines present chaotic dynamical behaviour of a very well-known type. This behaviour is generated by a dynamical feedback loop which also generates Julia sets.

MSC:

57M30 Wild embeddings
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] Blanchard ( P. ) .- Complex analytic dynamics on the Riemann sphere , B.A.M.S. 11 , n^\circ 1 ( 1984 ), pp. 85 - 141 . Article | MR 741725 | Zbl 0558.58017 · Zbl 0558.58017
[2] Feigenbaum ( M. ) . - Universal behaviour in nonlinear systems , Los Alamos Science 1 ( 1980 ), pp. 4 - 27 . MR 608366
[3] Freedman ( M. ) and Luo ( F. ) .- Selected applications of Geometry to low-dimensional Topology , Marker Lectures in the Math. Sciences , Princ. Univ. ( 1989 ). MR 1043633 | Zbl 0691.57001 · Zbl 0691.57001
[4] Poénaru ( V. ) . - Infinite processes and the 3-dimensional Poincaré conjecture 1 - The collapsible pseudo-spine representation theorem , Topology 31 ( 1992 ), pp. 625 - 636 . MR 1174263 | Zbl 0777.57013 · Zbl 0777.57013
[5] Poénaru ( V. ) .- Representations of open simply connected 3-manifolds, a finiteness result , Prépublication d’Orsay ( 1992 ). · Zbl 1052.57017
[6] Rolfsen ( D. ) .- Knots and Links , Math. Lect. Series ( 1976 ). MR 515288 | Zbl 0339.55004 · Zbl 0339.55004
[7] Smale ( S. ) .- Differentiable dynamical systems , Bull. A.M.S. 73 , n^\circ 6 ( 1967 ), pp. 747 - 817 . Article | MR 228014 | Zbl 0202.55202 · Zbl 0202.55202
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.