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\TopicFR{Systèmes dynamiques}
\TopicEN{Dynamical systems}

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\title{Corrigendum to ``The critical exponent functions''}

\alttitle{Corrigendum de \og Les fonctions de l'exposant critique \fg}

\author{\firstname{Dario} \lastname{Corona}}
\address{School of Science and Technology, University of Camerino, Italy}
\email{dario.corona@unicam.it}

\author{\firstname{Alessandro} \lastname{Della Corte}}
\address[1]{School of Science and Technology, University of Camerino, Italy}
\email{alessandro.dellacorte@unicam.it}

\Relation[corrige]{10.5802/crmath.286}
\CDRsetmeta{articletype}{erratum}

\begin{abstract}
We give a corrigendum to our paper~\cite{Critical} entitled ``The critical exponent functions''.
\end{abstract}

\begin{altabstract}
Nous donnons un corrigendum à notre article~\cite{Critical} intitulé \og Les fonctions de l'exposant critique \fg.
\end{altabstract}

\begin{document}

\maketitle

In~\cite{Critical}, a family of interval maps based on the word-combinatorial concept of critical exponent was introduced, hence referred to as ``critical exponent functions''. A purely combinatorial result, namely~\cite[Theorem~7]{Critical}, was presented and employed in that paper to prove some analytical and dynamical properties of such interval maps. However, \cite[Theorem~7]{Critical} does not hold. Denoting by $\{0,1\}^*$ and $\{0,1\}^\omega$ the sets of finite and infinite binary words, respectively, and using $E(w)$ to represent the critical exponent of a word, \cite[Theorem~7]{Critical} was formulated as follows: \emph{``Let $w \in \{0,1\}^*$. For every $\alpha > \max\bigl\{2,E(w)\bigr\}$, there exists $y \in \{0,1\}^\omega$ such that $E(wy) = \alpha$''}. This claim is false, as shown for instance by the counterexample $w=00100100$. Indeed, in this case we have $E(w)=8/3$, while both $w0$ and $w1$ are 3-powers, so that $E(wy)\ge 3 > 8/3$ for every $y\in\{0,1\}^\omega$.

However, most of the properties obtained through that combinatorial result remain valid with minor adjustments. Indeed, after reviewing all the properties of $\kappa_2$, we find that all of them hold, except for the following ones:
\begin{itemize}
\item The property of being left-\ or right-Darboux at every point, as stated in~\cite[Proposition~19]{Critical}; it remains unclear if this property holds or not.
\item The characterization of the almost fixed points of $\kappa_2$, specifically in~\cite[Corollary~24]{Critical}, and the subsequent uncountability of them (cf.~\cite[Corollary~25]{Critical}).
\item The existence of the fixed point can be granted only on any left-neighbourhood of $x_\tau$, not in a neighbourhood of any almost fixed point (cf.~\cite[Corollary~26]{Critical}); however, this suffices to prove the existence of countably many fixed points, so that~\cite[Corollary~27]{Critical} holds.
\end{itemize}

Analogously, most of the properties of $\kappa_n$ hold. The main results that fail are:
\begin{enumerate}
\item being left- or right-Darboux at every point (cf.~\cite[Corollary~42]{Critical});
\item the characterization of the almost-fixed points.
\end{enumerate}
All the properties presented in~\cite{Critical} for $\kappa$ hold.

The interested reader can find more details in~\cite{corona2024-critExp}.

\section*{Acknowledgments}

We are grateful to Sa\'ul Rodr\'iguez Mart\'in for pointing out the error.

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations.

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