A note on stronger forms of sensitivity for dynamical systems

Risong Li

Chaos, Solitons & Fractals, 2012, vol. 45, issue 6, 753-758

Abstract: Let (X,d) be a compact metric space and (κ(X),dH) be the space of all non-empty compact subsets of X equipped with the Hausdorff metric dH. The dynamical system (X,f) induces another dynamical system (κ(X),f¯), where f:X→X is a continuous map and f¯:κ(X)→κ(X) is defined by f¯(A)={f(a):a∈A} for any A∈κ(X). In this paper, we introduce the notion of ergodic sensitivity which is a stronger form of sensitivity, and present some sufficient conditions for a dynamical system (X,f) to be ergodically sensitive. Also, it is shown that f¯ is syndetically sensitive (resp. multi-sensitive) if and only if f is syndetically sensitive (resp. multi-sensitive). As applications of our results, several examples are given. In particular, it is shown that if a continuous map of a compact metric space is chaotic in the sense of Devaney, then it is ergodically sensitive. Our results improve and extend some existing ones.

Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:45:y:2012:i:6:p:753-758

DOI: 10.1016/j.chaos.2012.02.003

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