On topology of manifolds admitting gradient-like calscades with surface dynamics and on growth of the number of non-compact heteroclinic curves

We consider a class $GSD(M^3)$ of gradient-like diffeomorphisms with surface dynamics given on closed oriented manifold $M^3$ of dimension three. Earlier it was proved that manifolds admitting such diffeomorphisms are mapping tori under closed orientable surface of genus $g$, and the number of non-compact heteroclinic curves of such diffeomorphisms is not less than $12g$. In this paper, we determine a class of diffeomorphisms $GSDR (M^3) \subset GSD(M^3)$ that have the minimum number of heteroclinic curves for a given number of periodic points, and prove that the supporting manifold of such diffeomorphisms is a Seifert manifold. The separatrices of periodic points of diffeomorphisms from the class $ GSDR (M^3) $ have regular asymptotic behavior, in particular, their closures are locally flat. We provide sufficient conditions (independent on dynamics) for mapping torus to be Seifert. At the same time, the paper establishes that for any fixed $g \ geq 1$, fixed number of periodic points, and any integer $ n \geq 12g$, there exists a manifold $M^3$ and a diffeomorphism $f \in GSD (M^3)$ having exactly $ n $ non-compact heteroclinic curves.