  In this paper, for r in N with r>=2 we consider several stronger version
r-sensitivities and measure-theoretical r-sensitivities by analysing subsets of
nonnegative integers, for which the r-sensitivity occurs. We obtain an
Auslander-Yorke's type dichotomy theorem: a minimal topological dynamical
system is either thickly r-sensitive or an almost m to one extension of its
maximal equicontinuousfactor for some m in {1,2,...,r-1}.
