Avoid common mistakes on your manuscript.
Correction to: J. Fixed Point Theory Appl. (2023) 25:15 https://doi.org/10.1007/s11784-022-01037-5
Lemmas 5.4 and 10.4 in the original article are wrong. The first and the third paragraphs of the proof of Lemma 5.4 are correct, hence the largest f-invariant subset of E exists and is contained in \(I_f(E)\). However, the second paragraph is incorrect. In fact, the following counterexample exhibits that the subset \(I_f(E)\) is not necessarily f-invariant:
Let \(X = (\mathbb {N}\times \mathbb {N}) / {\sim }\), where the equivalence relation \(\sim \) is defined by
Define a self-map \(f :X \rightarrow X\) by \(f \overline{(a,b)} = \overline{(a+1,b)}\). Thus, f can be described by the following diagram:
Put \(E = X\). We see that
and hence
Thus, we have \(I_f(E) \ne f(I_f(E))\), i.e. \(I_f(E)\) is not f-invariant.
FormalPara Remark 2It is obvious that \(I_f(E)\) is a subset of E satisfying \(I_f(E) \subset {{{\,\textrm{Dom}\,}}} f\) and \(f(I_f(E)) \subset I_f(E)\). The issue is that \(I_f(E) \subset f(I_f(E))\) may not hold.
There is an analogous counterexample to Lemma 10.4:
Put \(X = (\mathbb {R}_{\geqslant 0} \times \mathbb {R}_{\geqslant 0}) / {\sim }\), where \(\sim \) is the equivalence relation defined by
Define a semiflow \(F :\mathbb {R}_{\geqslant 0} \times X \rightarrow X\) by \(F(t, \overline{(a,b)}) = \overline{(a+t,b)}\). Put \(E = X\). Then, \(I_F(E)\) is not F-invariant.
Fortunately, the failures of Lemmas 5.4 and 10.4 do not affect the other parts of the original article. This is because the following two weaker lemmas, which are enough for our purpose, hold true:
Let \(f :X \rightharpoonup X\) be a continuous partial self-map on a locally compact Hausdorff space X. Let K be a compact subset of \({{\,\textrm{Dom}\,}}f\). Then, \(I_f(K)\) is the largest f-invariant subset of K.
Proof It suffices to verify that \(I_f(K) \subset f(I_f(K))\). In other words, it is enough to see that the intersection \(f^{-1}(x) \cap I_f(K)\) is nonempty for any \(x \in I_f(K)\). Observe that, for each \(a,b \in \mathbb {N}\),
is a closed subset of K. Furthermore, it is nonempty since
From Remark 5.3 in the original article and the compactness of K, we conclude that
\(\square \)
Let \(F :\mathbb {R}_{\geqslant 0} \times X \rightharpoonup X\) be a continuous partial semiflow on a locally compact Hausdorff space X. Let K be a compact subset of X such that \([0, \varepsilon ] \times K \subset {{\,\textrm{Dom}\,}}F\) for some \(\varepsilon \in \mathbb {R}_{> 0}\). Then, \(I_F(K)\) is the largest F-invariant subset of K.
FormalPara ProofUsing Lemma 9.2 in the original article and arguing as in the proof of Lemma 4, we see that \(I_f(K) \subset f^t(I_F(K))\) (and hence, \(I_f(K) = f^t(I_F(K))\)) holds for \(t \in [0, \varepsilon ]\). This implies \(I_f(K) = f^t(I_F(K))\) for any \(t \in \mathbb {R}_{\geqslant 0}\). \(\square \)
We finally remark that there is another important situation in which Lemmas 5.4 and 10.4 are true:
We say that a partial map \(f :X \rightharpoonup Y\) is injective if it is injective as a map from \({{\,\textrm{Dom}\,}}X\) to Y.
FormalPara Lemma 7Let \(f :X \rightharpoonup X\) be an injective partial self-map on a set X. Let E be a subset of X. Then, \(I_f(E)\) is the largest f-invariant subset of E.
FormalPara ProofIf f is injective, the second paragraph of the proof of Lemma 5.4 in the original article is valid as written; direct images under injective maps commute with intersections. \(\square \)
FormalPara Lemma 8Let \(F :\mathbb {R}_{\geqslant 0} \times X \rightharpoonup X\) be a partial semiflow on a set X such that \(f^t :X \rightharpoonup X\) is injective for any \(t \in \mathbb {R}_{\geqslant 0}\). Let E be a subset of X. Then, \(I_F(E)\) is the largest F-invariant subset of E.
FormalPara ProofThe same as the proof of Lemma 7. \(\square \)
Acknowledgements
The author was supported by JSPS KAKENHI Grant Number 19K14529.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Morita, Y. Correction to: Conley index theory without index pairs. I: The point-set level theory. J. Fixed Point Theory Appl. 26, 2 (2024). https://doi.org/10.1007/s11784-023-01094-4
Accepted:
Published:
DOI: https://doi.org/10.1007/s11784-023-01094-4