Diagonals separating the square of a continuum

A metric continuum $X$ is indecomposable if it cannot be put as the union of two of its proper subcontinua. A subset $R$ of $X$ is said to be continuumwise connected provided that for each pair of points $p,q\in R$, there exists a subcontinuum $M$ of $X$ such that $\{p,q\}\subset M\subset R$. Let $X^2$ denote the Cartesian square of $X$ and $\mathrm{\Delta }$ the diagonal of $X^2$. In \cite{ka} it was asked if for a continuum $X$, distinct from the arc, $X^2\setminus \mathrm{\Delta }$ is continuumwise connected if and only if $X$ is decomposable. In this paper we show that no implication in this question holds. For the proof of the non-necessity, we use the dynamic properties of a suitable homeomorphism of the Cantor set onto itself to construct an appropriate indecomposable continuum $X$.