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A type of moduli for saddle connections of planar diffeomorphisms. (English) Zbl 0672.58036
The authors consider diffeomorphisms on a 2-dimensional manifold having a pair of hyperbolic fixed points p and q of saddle type so that one of the connected components of \(W^ u(p)-\{p\}\) is a connected component of \(W^ s(q)-\{q\}\). They are interested in a classification of such situation up to conjugacies outside this common separatrice of p, q. Such a conjugacy gives rise to a genuine conjugacy after pinching down this separatrice to a point. An invariant, called “the transition function”, was associated with such a pair of hyperbolic points in a paper by the first author [Topology 19, 9-21 (1980; Zbl 0447.58025)]. Here the authors prove that if the transition function is strictly monotone, then there are no moduli for the above type of conjugacy. If the transition function is piecewise strictly monotone but not monotone, then the so-called Palis index [cf. J. Palis, Astérisque 51, 335-346 (1978; Zbl 0396.58015)] is proved to be an invariant of this type (“pinched”) of conjugacy.
Reviewer: N.V.Ivanov

MSC:
37D99 Dynamical systems with hyperbolic behavior
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[1] Camacho, M.I, Geometric properties of homogeneous vector fields of degree two in \(R\)^{3}, Trans. amer. math. soc., 268, No. 1, 79-101, (1981) · Zbl 0484.58020
[2] {\scF. Dumortier, P. R. Rodrigues, and R. Roussarie}, Germs of diffeomorphisms in the plane, in “Lecture Notes in Mathematics,” Vol. 902, Springer-Verlag, Berlin/New York. · Zbl 0502.58001
[3] Hartman, P, On local homeomorphisms of Euclidean spaces, Bol. soc. mat. mexicana, 5, (1960) · Zbl 0127.30202
[4] Hirsch, M; Pugh, C.C, Stable manifolds and hyperbolic sets, (), 133-163 · Zbl 0215.53001
[5] de Melo, W, Moduli of stability of two-dimensional diffeomorphisms, Topology, 19, 9-21, (1980) · Zbl 0447.58025
[6] de Melo, W; van Strien, S.J, Diffeomorphisms on surfaces with a finite number of moduli, Ergotic theory dynamical systems, 7, 415-462, (1987) · Zbl 0609.58023
[7] Newhouse, S; Palis, J; Takens, F, Bifurcation and stability of families of diffeomorphisms, Publ. math. I.H.E.S., 57, (1983) · Zbl 0518.58031
[8] Palis, J, A differentiable invariant of topological conjugacies and moduli of stability, Astérisque, 51, 335-346, (1978) · Zbl 0396.58015
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