Special homeomorphisms and approximation for Cantor systems

E. Akin, E. Glasner, and B. Weiss had constructed the special homeomorphism that has a dense Gδ conjugacy class in the space of all Cantor homeomorphisms. M. Hochman showed that the universal odometer is the special homeomorphism in the space of all topologically transitive Cantor homeomorphism. Following the approach of E. Akin, E. Glasner, and B. Weiss, we show that the universal odometer is the special homeomorphism in the space of all chain transitive Cantor systems. We extend this result to the space of chain transitive systems that are restricted by a periodic spectrum. Further, we construct the special homeomorphism in the space of all chain recurrent systems. In doing so, every 0-dimensional system is described as the inverse limit of a sequence of finite directed graphs and graph homomorphisms. In the previous paper, we had shown that a certain periodic condition determines whether a Cantor system approximates a chain mixing Cantor system by topological conjugacies. We shall extend this result to the chain transitive case. These conditions are described in terms of sequences of finite directed graphs and graph homomorphisms.