SENSITIVITY, PROXIMAL EXTENSION AND HIGHER ORDER ALMOST AUTOMORPHY.

Let (X, T) be a topological dynamical system, and ℱ be a family of subsets of ℤ+. (X, T) is strongly ℱ-sensitive if there is δ > 0 such that for each non-empty open subset U there are x, y ∈ U with {n ∈ ℤ+ : d(Tnx, Tny) > δ} ∈ ℱ. Let ℱt (resp. ℱip, ℱfip) consist of thick sets (resp. IP-sets, subsets containing arbitrarily long finite IP-sets). The following Auslander-Yorke’s type dichotomy theorems are obtained: (1) a minimal system is either strongly ℱip-sensitive or an almost one-to-one extension of its ∞-step nilfactor; (2) a minimal system is either strongly ℱip-sensitive or an almost one-to-one extension of its maximal distal factor; (3) a minimal system is either strongly ℱt-sensitive or a proximal extension of its maximal distal factor. [ABSTRACT FROM AUTHOR]