In [Reference GutmanGut17, Theorem B.3] there is a mismatch between the statement of the theorem and its proof. In order to correct the situation it is enough to replace the expression ‘isomorphic extension’ in the statement of Theorem B.3 and in its proof by the expression ‘strongly isomorphic extension’. We proceed to give the necessary background.

We consider topological dynamical systems $(X,T)$ where X is compact and metric and $T\colon X \to X$ is a homeomorphism. Recall that a Borel subset of X is called a full set if it has measure $1$ with respect to any T-invariant Borel probability measure on X.

Recall that an extension $\pi \colon (X,T)\to (Y,S)$ is called strongly isomorphic if there is a full set $E\subset Y$ such that the restriction of $\pi $ to $\pi ^{-1}(E)$ is one-to-one [Reference BurguetBur19, §2.3]. This condition is easily seen to be equivalent to the condition that the Borel set

$$ \begin{align*} \{y\in Y:\ |\pi^{-1}(y)|=1\} \end{align*} $$

is full (note that the set $\{y\in Y:\ |\pi ^{-1}(y)|=1\}=\bigcap _{n=1}^{\infty }\{y\in Y:\ \operatorname {diam}(\pi ^{-1}(y))<1/n\}$ is $G_\delta $ ).

We remark that the equivalence between the small boundary property and the existence of a zero-dimensional strongly isomorphic extension is proven in [Reference Kerr and SzabóKS20, Theorem 5.5] using a different terminology. It is an open problem whether the existence of a zero-dimensional isomorphic extension implies the existence of a zero-dimensional strongly isomorphic extension.