Not finitely but countably Hopf-equivalent clopen sets in a Cantor minimal system (Q1039212)

\textit{E. Hopf}'s notion [Trans. Am. Math. Soc. 34, 373--393 (1932; Zbl 0004.26001)] of equivalence for non-singular transformations in the measurable category characterizes the existence of an equivalent finite invariant measure, and the conservative property of the map, in terms of countable and finite Hopf equivalence, respectively. Hopf equivalence was later extended to the topological setting, and played a role in the topological orbit-equivalence classification of Cantor minimal systems. In this paper the author continues his work [Japan J. Math. 28, 299--312 (2002; Zbl 1038.37009)] on topological Hopf equivalence in zero-dimensional systems by finding an example of clopen sets in a Cantor minimal system which are countably Hopf equivalent but not finitely Hopf equivalent. The proof uses the Bratelli--Versik representation of the minimal shift arising from the Morse substitution. Using the orbit equivalence theory of \textit{T. Giordano, I. F. Putnam} and \textit{C. F. Skau} [J. Reine Angew. Math. 469, 51--111 (1995; Zbl 0834.46053)] the orbit structure of a Cantor minimal system is characterized in terms of Hopf equivalences.