In this paper we give examples of \(C^{2+\delta}\) diffeomorphisms of the annulus A permuting a countable set of disjoint disks \(\{R_ n\}\subset A\). Theorem A. For \(\delta >0\) sufficiently small there exists a Jordan curve \(Q\subset A\), a family of disjoint disks \(\{R_ n\}\subset A\) with \(R_ n\cap Q\neq \emptyset\) and a \(C^{2+\delta}\) diffeomorphism f: \(A\to A\) such that \(Q\cup \{R_ n\}\) is f-invariant and has no periodic points. The curve Q has Hausdorff dimension \(1+\delta\). The derivative of f at the minimal set in Q is an isometry, a feature shared by the canonical Denjoy counterexample D. This property is useful in the author's paper in Topology 27, No.3, 249-278 (1988), where \(C^{2+\delta}\) counterexamples to the Seifert conjecture are found. There may be periodic points of f in a neighborhood of \(Q\cup \{R_ n\}\). In the mentioned paper f is made to be semi-stable so there is no longer any periodicity. An overview of this paper and its sequel [loc. cit.] may be found in the author's paper in Bull. Am. Math. Soc., New Ser. 13, 147-153 (1985; Zbl 0587.58039).