Minimal subsystems of triangular maps of type 2∞; Conclusion of the Sharkovsky classification program

The subject of this paper is to give the description, up to topological conjugacy, of possible minimal sets of triangular maps of the square of type 2∞. In [4], we give a general method allowing to embed any zero-dimensional almost 1–1 extension of the dyadic odometer (in particular any dyadic Toeplitz system) as a minimal set of a triangular map of this type. In this paper we present a method (a combination of that described in [4] with one introduced in [1]) of similarly embedding a special class of zero-dimensional almost 2–1 extensions of the odometer. We conjecture that these two embedding theorems exhaust all possibilities for nonperiodic minimal sets.The paper was inspired by the last unsolved problem in the Sharkovski classification program of triangular maps: does there exist a triangular map with positive entropy attained on the set of uniformly recurrent points but with entropy zero on the set of regularly recurrent points. The paper answers this question positively, concluding the program.