Trellises formed by stable and unstable manifolds in the plane

Easton, Robert W.

doi:10.1090/S0002-9947-1986-0825732-X

Trellises formed by stable and unstable manifolds in the plane
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by Robert W. Easton

Trans. Amer. Math. Soc. 294 (1986), 719-732

DOI: https://doi.org/10.1090/S0002-9947-1986-0825732-X

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Abstract:

A trellis is the figure formed by the stable and unstable manifolds of a hyperbolic periodic point of a diffeomorphism of a $2$-manifold. This paper describes and classifies some trellises. The set of homoclinic points is linearly ordered as a subset of the stable manifold and again as a subset of the unstable manifold. Each homoclinic point is assigned a type number which is constant on its orbit. Combinatorial properties of trellises are studied using type numbers and the pair of linear orderings. Trellises are important because their closures in some cases are strange attractors and in other cases are ergodic zones.

References

Roy L. Adler and Benjamin Weiss, Similarity of automorphisms of the torus, Memoirs of the American Mathematical Society, No. 98, American Mathematical Society, Providence, R.I., 1970. MR 0257315
Marcy Barge, Horseshoe maps and inverse limits, Pacific J. Math. 121 (1986), no. 1, 29–39. MR 815029
Jürgen Moser, Stable and random motions in dynamical systems, Annals of Mathematics Studies, No. 77, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1973. With special emphasis on celestial mechanics; Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J. MR 0442980
Stephen Smale, Diffeomorphisms with many periodic points, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 63–80. MR 0182020