proceedings of the american mathematical society Volume 118, Number 4, August 1993 CHAIN TRANSITIVITY AND ROTATION SHADOWING FOR ANNULUS HOMEOMORPHISMS FERNANDA BOTELHO AND LIANG CHEN (Communicated by Charles Pugh) Abstract. We present a relation between the rotation of chain transitive sets and the rotation shadowing for annulus homeomorphisms isotopic to identity. 1. Introduction The concept of rotation shadowing was introduced by Barge and Swanson in [BS]. They conjectured that the rotation shadowing property is generic for degree one annulus homeomorphisms. Roughly, rotation shadowing means that for small 8 > 0 the rotation averages along a f5-pseudo-orbit can be uniformly approximated by the rotation averages along a true orbit; see §2 for the precise definition. In practice, rotation shadowing reveals whether computer calcula- tions of rotation numbers are accurate. In this paper we study the relationship between rotation shadowing and ro- tations of chain transitive sets for annulus homeomorphisms. Our main result is the following Theorem. Suppose A is a compact annulus and /: A —> A is a homeomorphism isotopic to identity. The rotation number is well defined on every chain transitive set in A if and only if f has the rotation shadowing property and each point of A has a well-defined rotation number. If the rotation set of an annulus homeomorphism is nowhere dense, [Sw] proves that the rotation number is well defined and varies continuously over chain recurrent sets in A. In such a case, the results of [Fr, Ha] show that the rotation number is also well defined on every chain transitive set (see [BC]). Consequently, the theorem implies that, under the same assumption, the annu- lus homeomorphism has the rotation shadowing property. 2. Definitions Let d be the Euclidean metric in a compact annulus A, A = R x [0, 1 ] be the universal covering space of A, and p be the standard covering map. If / Received by the editors December 27, 1991. 1991 Mathematics Subject Classification. Primary 34C35; Secondary 54H20. Key words and phrases. Chain transitivity, shadowing, pseudo-orbit, rotation set, annulus homeo- morphisms. ©1993 American Mathematical Society 0002-9939/93 $1.00+ $.25 per page 1173 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use