On topological classification of regular Denjoy type homeomorphisms

We consider regular Denjoy type homeomorphisms of the two-dimensional torus which are the most natural generalization of Denjoy homeomorphisms of the circle. In particular, they arise as Poincaré maps induced on global cross sections by leaves of one-dimensional orientable unstable foliations of some partially hyperbolic diffeomorphisms of closed three-dimensional manifolds. 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 on the two-dimensional torus. For regular Denjoy type homeomorphisms, we introduce a complete invariant of topological conjugacy characterized by the minimal translation, which is semiconjugate to the given regular Denjoy type homeomorphism, with a distinguished at most countable set of orbits.