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Correction to: Devaney’s and Li-Yorke’s Chaos in Uniform Spaces

  • Tatsuya AraiEmail author
Correction
  • 34 Downloads

Correction to: J Dyn Control Syst

 https://doi.org/10.1007/s10883-017-9360-0

The original version of this article unfortunately contained a mistake.

In the second paragraph of Definition 2.4, we defined elements x and y to be asymptotic if for any \(U\in \mathcal {U}\) and any \(\{G_{i}|i \in \mathbb {N}\} \in \mathcal {G}\), there exists \(k \in \mathbb {N}\) such that (gx,gy) ∈ U for each gGGk. However, this definition is incorrect, because it holds that x and y are asymptotic if and only if x = y. Thus the second paragraph of Definition 2.4 needs to be corrected as follows:

Elements x and y of X to be asymptotic if for any \(U \in \mathcal {U}\) and any \(\{G_{i} |i \in \mathbb {N}\}\in \mathcal {G}\), there exists \(k \in \mathbb {N}\) such that (gx,gy) ∈ U for each \(g\in \bigcup _{i\in \mathbb {N}} G_{i} \backslash G_{k} \).

And the set AR of the forth paragraph of Definition 2.4 should be denoted by
$$AR=\bigcap\limits_{U\in \mathcal{U}} {\bigcap\limits_{\{G_{i} \}\in \mathcal{G}}{\bigcup\limits_{k\in \mathbb{N}} \bigcap\limits_{g\in \bigcup {_{i\in \mathbb{N}} G_{i} \backslash G_{k} } } {\left( {g\times g} \right)} } }^{-1}\text{Cl}_{X} \left( U \right). $$
Along with the above corrections, we need to correct as follows:
  • The phrase “let G be an Abelian group” in the statement of Theorem 1.2 is replaced by “let G be a countable Abelian group”. Similarly, the phrases “an Abelian group G” in Lemma 3.6 and Proposition 3.7 are replaced by “a countable Abelian group G”.

  • In the proof of Lemma 3.6, the phrase “Let \((G_{i})_{i\in \mathbb {N}}\) be an elements of \(\mathcal {G}\) with G1 = ” of the line 1 in the second paragraph should be replaced by “Let \((G_{i})_{i\in \mathbb {N}}\) be an elements of \(\mathcal {G}\) with G1 = and \(G=\bigcup _{i\in \mathbb {N}} G_{i} \)”.

The author is very grateful to Fatemah Ayatollah Zadeh Shirazi for pointing out the mistake and providing useful suggestions.

Notes

References

  1. 1.
    Arai T. Devaney’s and Li-Yorke’s chaos in uniform spaces. J Dyn Control Syst 2018;24(1):93–100.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Tsukuba University of TechnologyIbarakiJapan

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