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

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 g ∈ G∖G_k. 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.