Stronger forms of sensitivity for dynamical systems. (English) Zbl 1132.54023

Moothathu, T. K. Subrahmonian

Nonlinearity 20, No. 9, 2115-2126 (2007).

Discrete dynamical systems induced by continuous self-maps of compact metric spaces are considered. The paper considers the problem of dependence on initial conditions from the frame of “large” subsets of \(\mathbb{N}\). Mainly considered are “syndetic sensitivity” and “cofinite sensitivity”. In this setting the following is established:
(i) Any syndetically transitive, non-minimal map is syndetically sensitive;
(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) A transitive, sensitive map and not syndetically sensitive is constructed.

Reviewer: Juan Luis García Guirao (Cartagena)

MSC:

54H20	Topological dynamics (MSC2010)
37B10	Symbolic dynamics