Geodesic
Moduli of $\Omega$-conjugacy of two-dimensional diffeomorphisms with a structurally unstable heteroclinic contour
Sbornik. Mathematics, Tome 187 (1996) no. 9, pp. 1261-1281 
S. V. Gonchenko. Moduli of $\Omega$-conjugacy of two-dimensional diffeomorphisms with a structurally unstable heteroclinic contour. Sbornik. Mathematics, Tome 187 (1996) no. 9, pp. 1261-1281. http://geodesic.mathdoc.fr/item/SM_1996_187_9_a0/
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     title = {Moduli of $\Omega$-conjugacy of two-dimensional diffeomorphisms with a~structurally unstable heteroclinic contour},
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In this paper we consider two-dimensional diffeomorphisms with a structurally unstable heteroclinic contour consisting of two saddle fixed points and two heteroclinic trajectories: a structurally stable one and a structurally unstable one. Such diffeomorphisms are divided into three classes, depending on the structure of the set $N$ of trajectories lying entirely in a neighbourhood of the contour. For diffeomorphisms of the first and the second classes $N$ can be fully described. We show that the diffeomorphisms of the third class have $\Omega$-moduli, which are continuous topological conjugacy invariants on the set of non-wandering trajectories. We explicitly show two such moduli: $\theta$ and $\tau_0$. We discuss sufficient conditions of $\Omega$-conjugacy for rational $\theta$ and we also prove that on the bifurcation surface of diffeomorphisms of the third class the systems with a denumerable set of $\Omega$-moduli are dense.

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