Abstract
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 of trajectories lying entirely in a neighbourhood of the contour. For diffeomorphisms of the first and the second classes can be fully described. We show that the diffeomorphisms of the third class have -moduli, which are continuous topological conjugacy invariants on the set of non-wandering trajectories. We explicitly show two such moduli: and . We discuss sufficient conditions of -conjugacy for rational and we also prove that on the bifurcation surface of diffeomorphisms of the third class the systems with a denumerable set of -moduli are dense.
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