A note on ℱ-sensitivity for non-autonomous systems

ABSTRACT

We study some stronger forms of sensitivity, namely, $\mathcal{F}$-sensitivity and weakly $\mathcal{F}$-sensitivity for non-autonomous discrete dynamical systems. We obtain a condition under which these two forms of sensitivity are equivalent. We also justify the difference between $\mathcal{F}$-sensitivity and some other stronger forms of sensitivity through examples. We explore the relation between the $\mathcal{F}$-sensitivity of the non-autonomous system $(X,f_{1,\mathrm{\infty }})$ and autonomous system $(X,f)$, where $f_n$ is a sequence of continuous functions converging uniformly to f. We also study the $\mathcal{F}$-sensitivity of a non-autonomous system $(X,f_{1,\mathrm{\infty }})$, generated by a finite family of maps $\mathbb{F}=\{f_1,f_2,\dots ,f_k\}$ and give an example showing that such non-autonomous systems can be $\mathcal{F}$-sensitive, even when none of the maps in the family $\mathbb{F}$ is $\mathcal{F}$-sensitive.

KEYWORDS:

• Non-autonomous dynamical system
• Furstenberg families
• $\mathcal{F}$-sensitive
• weakly $\mathcal{F}$-sensitive

MSC (2010) CLASSIFICATIONS:

• Primary: 54H20
• Secondary: 37B55

Acknowledgments

The first author is funded by the Government of India, Ministry of Science and Technology, no. DST/INSPIRE Fellowship/[IF160750]. The authors are thankful to the referee for his/her valuable suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.