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06-03-2019
Stronger Forms of Transitivity and Sensitivity for Nonautonomous Discrete Dynamical Systems and Furstenberg Families
Published in: Journal of Dynamical and Control Systems | Issue 1/2020
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Abstract
Let (Y, d) be a nontrivial metric space and (Y, g1,∞) be a nonautonomous discrete dynamical system given by sequences \((g_{l})_{l = 1}^{\infty }\) of continuous maps gl : Y → Y and let \(\mathcal {F}\), \(\mathcal {F}_{1}\) and \(\mathcal {F}_{2}\) be given shift-invariant Furstenberg families. In this paper, we study stronger forms of transitivity and sensitivity for nonautonomous discrete dynamical systems by using Furstenberg family. In particular, we discuss the \(\mathcal {F}\)-transitivity, \(\mathcal {F}\)-mixing, \(\mathcal {F}\)-sensitivity, \(\mathcal {F}\)-collective sensitivity, \(\mathcal {F}\)-synchronous sensitivity, \((\mathcal {F}_{1},\mathcal {F}_{2})\)-sensitivity and \(\mathcal {F}\)-multi-sensitivity for the system (Y, g1,∞) and show that under the conditions that gj is semi-open and satisfies gj ∘ g = g ∘ gj for each j ∈ {1, 2, ⋯ } and that
exists (i.e., \(\sum \limits _{j = 1}^{\infty }D(g_{j},g)<+\infty \)), the following hold:
$$\sum\limits_{j = 1}^{\infty}D(g_{j},g) $$
(1)
(Y, g1,∞) is \(\mathcal {F}\)-transitive if and only if so is (Y, g).
(2)
(Y, g1,∞) is \(\mathcal {F}\)-mixing if and only if so is (Y, g).
(3)
(Y, g1,∞) is \(\mathcal {F}\)-sensitive if and only if so is (Y, g).
(4)
(Y, g1,∞) is \((\mathcal {F}_{1},\mathcal {F}_{2})\)-sensitive if and only if so is (Y, g).
(5)
(Y, g1,∞) is \(\mathcal {F}\)-collectively sensitive if and only if so is (Y, g).
(6)
(Y, g1,∞) is \(\mathcal {F}\)-synchronous sensitive if and only if so is (Y, g).
(7)
(Y, g1,∞) is \(\mathcal {F}\)-multi-sensitive if and only if so is (Y, g).
The above results extend the existing ones.