On $2$-diffeomorphisms with one-dimensional basic sets and a finite number of moduli

This paper is a step towards the complete topological classification of $\Omega$-stable diffeomorphisms on an orientable closed surface, aiming to give necessary and sufficient conditions for two such diffeomorphisms to be topologically conjugate without assuming that the diffeomorphisms are necessarily close to each other. In this paper we will establish such a classification within a certain class $\Psi$ of $\Omega$-stable diffeomorphisms defined below. To determine whether two diffeomorphisms from this class $\Psi$ are topologically conjugate, we give (i) an algebraic description of the dynamics on their non-trivial basic sets, (ii) a geometric description of how invariant manifolds intersect, and (iii) define numerical invariants, called moduli, associated to orbits of tangency of stable and unstable manifolds of saddle periodic orbits. This description determines the scheme of a diffeomorphism, and we will show that two diffeomorphisms from $\Psi$ are topologically conjugate if and only if their schemes agree.