A type of moduli for saddle connections of planar diffeomorphisms.

*(English)*Zbl 0672.58036The 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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\textit{W. De Melo} and \textit{F. Dumortier}, J. Differ. Equations 75, No. 1, 88--102 (1988; Zbl 0672.58036)

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##### References:

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