Dmitrii Mints

In P.D.McSwiggen’s article, it was proposed Derived from Anosov type construction which leads to a partially hyperbolic diffeomorphism of the 3-torus. The nonwandering set of this diffeomorphism contains a two-dimensional attractor which consists of one-dimensional unstable manifolds of its points. The constructed diffeomorphism admits an invariant onedimensional orientable foliation such that it contains unstable manifolds of points of the attractor as its leaves. Moreover, this foliation has a global cross section (2-torus) and defines on it a Poincar´e map which is a regular Denjoy type homeomorphism. Such homeomorphisms are the most natural generalization of Denjoy homeomorphisms of the circle and play an important role in the description of the dynamics of aforementioned partially hyperbolic diffeomorphisms. In particular, the topological conjugacy of corresponding Poincaré maps provides necessary conditions for the topological conjugacy of the restrictions of such partially hyperbolic diffeomorphisms to their two-dimensional attractors. The nonwandering set of each regular Denjoy type homeomorphism is a Sierpiński set and each such homeomorphism is, by definition, semiconjugate to the minimal translation of the 2-torus. We introduce a complete invariant of topological conjugacy for regular Denjoy type homeomorphisms that is characterized by the minimal translation, which is semiconjugation of the given regular Denjoy type homeomorphism, with a distinguished, no more than countable set of orbits.

This paper is a continuation of our previous work where we investigated the class $\mathbb G(M^2)$ of $A$-diffeomorphisms of closed orientable connected surfaces such that their nonwandering sets consist of one-dimensional basic sets (attractors and repellers). In that work, we showed that the dynamical properties of each diffeomorphism from a given class define a collection consisting of nonempty multisets of natural numbers (each such collection contains at least two multisets). These multisets are topological invariants of the diffeomorphism and uniquely determine the topology of the ambient surface. In this paper, we solve the problem of realization of diffeomorphisms from the class $\mathbb G(M^2)$ with respect to a given collection of multisets of natural numbers. We describe all possible collections of multisets from which one can construct a diffeomorphism from the class $\mathbb G(M^2)$, presenting a step-by-step algorithm of construction.