A homeomorphism \(T\) of a compact metric space \(X\) is called (topologically) coalescent if every continuous map which commutes with \(T\) is necessarily invertible. A Toeplitz sequence over a finite alphabet \(\Sigma\) is a sequence \(\omega\in \Sigma^{\mathbb{Z}}\) in which every entry is repeated periodically. More precisely, there exist positive integers \(p_n\) with \(p_n| p_{n+1}\) such that for every positive integer \(k\) there is an \(n\) for which \[ \omega(-k)\dots \omega(k)= \omega(-k+jp_n)\dots \omega(k+jp_n) \] for all \(j\in\mathbb{Z}\). The Toeplitz sequence \(\omega\) is said to be regular if the density of its \(p_n\)-periodic part tends to 1. A Toeplitz flow is a subshift of the full shift \((\Sigma^{\mathbb{Z}},S)\) defined as the orbit closure \((\overline{O} (\omega),S)\) of a Toeplitz sequence \(\omega\). Toeplitz flows are always minimal. In the regular case they are strictly ergodic and have discrete \(L^2\)-spectrum [K. Jacobs and M. Keane, Z. Wahrscheinlichkeitstheorie Verw. Geb. 13, 123-131 (1969; Zbl 0195.52703)]. Some classes of Toeplitz flows are known to consist solely of coalescent flows [T. Downarowicz, J. Kwiątkowski and Y. Lacroix, Colloq. Math. 68, No. 2, 219-228 (1995; Zbl 0820.28009)] and the question whether a noncoalescent one exists has been around for a few years.
The author presents an original construction of a noncoalescent regular Toeplitz flow based on an ingenious use of a \(\mathbb{Z}^2\)-action. In addition, the flow admits a commuting noninvertible continuous map for which the preimage of every Toeplitz sequence in \(\overline{O}(\omega)\) is a singleton.