Moduli of $\Omega$-conjugacy of two-dimensional diffeomorphisms with a structurally unstable heteroclinic contour

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.