Stronger forms of sensitivity for dynamical systems

Abstract

For continuous self-maps of compact metric spaces, we initiate a preliminary study of stronger forms of sensitivity formulated in terms of ‘large’ subsets of . Mainly we consider ‘syndetic sensitivity’ and ‘cofinite sensitivity’. We establish the following. (i) Any syndetically transitive, non-minimal map is syndetically sensitive (this improves the result that sensitivity is redundant in Devaney's definition of chaos). (ii) Any sensitive map of [0,1] is cofinitely sensitive. (iii) Any sensitive subshift of finite type is cofinitely sensitive. (iv) Any syndetically transitive, infinite subshift is syndetically sensitive. (v) No Sturmian subshift is cofinitely sensitive. (vi) We construct a transitive, sensitive map which is not syndetically sensitive.