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1. Introduction and definitions. We construct, using the lexicographic or "adic" maps of Vershik (as modified by Herman, Putnam, and Skau [P], [HPS], [Sk]), examples of minimal homeomorphisms of all possible (topological) entropies (including infinite) which are "strongly" orbit equivalent to the dyadic adding machine. In particular, this answers in a decisively negative way the question as to whether orbit equivalent homeomorphisms have the same entropy. We also show that any minimal homeomorphism of the Cantor set is strongly orbit equivalent to one of zero entropy. We recall some definitions. A source for these is the survey article of Skau [Sk]. A Bratteli diagram B is a directed graph whose vertex set decomposes into finite subsets, "levels" or "rows", Bk (k = 0,1,...), together with edges from vertices in Bk to vertices in Bk+X additionally, every vertex of Bk is joined to a vertex of Bk+X. The Bratteli diagram is simple if for all k, there exists k1> k such that for every vertex v in Bk and every vertex v' in Bk>, there is a path from v Xo v'. The Bratteli diagram is pointed if |2f o| = 1, that is, there is a distinguished top vertex. The set of infinite paths, usually denoted X, is called the Bratteli compactum, and is a compact zero

Entropy versus orbit equivalence for minimal homeomorphisms