Embedding Toeplitz systems in triangular maps: The last but one problem of the Sharkovsky classification program

Abstract

We give an example of a triangular map of the unit square containing a minimal Li–Yorke chaotic set and such that, in the whole system, there are no DC3-pairs. This solves the last but one problem of the Sharkovsky program of classification of triangular maps. We use completely new methods, in fact we show that every zero-dimensional almost 1–1 extension of the dyadic odometer can be realized as the unique nonperiodic minimal set in a triangular map of type 2∞. In case of a regular Toeplitz system we can additionally arrange that all invariant measures are supported by minimal sets.