A new notion of mild mixing is introduced for non-autonomous discrete systems, with proofs that mild mixing implies mixing and multi-transitivity, plus counterexamples correcting flawed theorems in the literature.
Iteration problem for several chaos in non-autonomous discrete system
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abstract
In this paper we investigate the iteration problem for several chaos in non-autonomous discrete system. Firstly, we prove that the Li-Yorke chaos of a non-autonomous discrete dynamical system is preserved under iterations when $f_{1,\infty}$ converges to $f$, which weakens the condition in the literature that $f_{1,\infty}$ uniformly converges to $f$. Besides, we prove that both DC2' and Kato's chaos of a non-autonomous discrete dynamical system are iteration invariants. Additionally, we give a sufficient condition for non-autonomous discrete dynamical system to be Li-Yorke chaos. Finally, we give an example to show that the DC3 of a non-autonomous discrete dynamical system is not inherited under iterations, which partly answers an open question proposed by Wu and Zhu(Chaos in a class of non-autonomous discrete systems, Appl.Math.Lett. 2013,26:431-436).
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A note on multi-transitivity in non-autonomous discrete systems
A new notion of mild mixing is introduced for non-autonomous discrete systems, with proofs that mild mixing implies mixing and multi-transitivity, plus counterexamples correcting flawed theorems in the literature.