A criterion for Toeplitz flows to be topologically isomorphic and applications

A dynamical system is said to be coalescent if its only endomorphisms are automorphisms. The question whether there exist coalescent ergodic dynamical systems with positive entropy has not been solved so far and it seems to be difficult. The analogous problem in topological dynamics has been solved by Walters ([W]). His example, however, is not minimal. In [B-K2], a class of strictly ergodic (hence minimal) Toeplitz flows is presented, which have positive entropy and trivial topological centralizers (the last condition implies coalescence). The entropy, however, is only estimated from below. Also the class is obtained in a not completely constructive way. The basic idea of this paper is contained in Section 2, in a criterion which describes homomorphisms (isomorphisms) between Toeplitz flows in terms of a block code simplified to a function sending blocks of a given length to blocks of the same length. This idea is then applied in Section 3 to effectively construct an uncountable family of pairwise nonisomorphic Toeplitz flows with topological entropy equal to a common arbitrarily preset value. Furthermore, all the Toeplitz flows have the same maximal uniformly continuous factor. In Section 4 we obtain conditions sufficient for coalescence of a Toeplitz flow. In particular, all Toeplitz flows of Section 3 turn out to be coalescent. We are grateful to Professor P. Liardet for several helpful conversations and valuable remarks.