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 N. 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.

Furstenberg families and chaos on uniform limit maps

“…Corollary 3 and Theorem 5 from [16] show that every Sturmian subshift is syndetically sensitive, and that no Sturmian subshift is cofinitely sensitive. In addition, Theorem 7 in [16] shows that there exists a sensitive subshift which is not syndetically sensitive. Consequently, there exist sensitive transformations that are not syndetically sensitive, and syndetically sensitive maps that are not cofinitely sensitive.…”

“…Therefore, when is a system sensitive? This question has gained some attention in more recent papers (see [1,4,8,9,15,16]). More recently, in [19] the authors gave an equivalence conditions for the uniform limit map f to be sensitive.…”

“…Especially, if this set is quite thin with arbitrarily large gaps between consecutive entries, then one has some excuse for treating the dynamical system as practically non-sensitive! For continuous self-maps of compact metric spaces, Moothathu [16] introduced stronger forms of sensitivity in terms of large subsets of Z + . Mainly he considered syndetic sensitivity and cofinite sensitivity.…”

Sensitivity of non-autonomous discrete dynamical systems revisited

“…For an ADS, Moothathu [10] initiated a preliminary study of stronger forms of sensitivity formulated in terms of some subsets of Z + , namely the syndetical sensitivity and cofinite sensitivity. Similarly to Moothathu [10], a NADS (X, {f n } ∞ n=1 ) is said to be: Banks et al [2] proved that every transitive ADS whose periodic points are dense in the state space has sensitive dependence on initial conditions. Based on this result, Lan [8] posed the following open problem (Problem 1.1).…”

F-sensitivity and (F_1, F_2)-sensitivity between dynamical systems and their induced hyperspace dynamical systems

“…Therefore, it is very important to study what system is sensitive. This problem has gained much attention recently (see [1,2,6,10,12,14,15,20,29,31]). There are several forms of sensitivity for dynamical systems (see [15]).…”