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For each pair of nonzero integers j, k, we define a homeomorphism fik of the two-disk minus three holes. We show that there exists a compact, invariant, hyperbolic set for each f1k on which fik iS conjugate to a subshift of finite type. This implies that the topological entropy of fkk is bounded below by 41jl Ikl 21l1 21kl. The object of this note is to describe some of the orbit structure of a class of homeomorphisms of the two-disk minus three holes. We call these mappings linked twist maps. They have arisen in several different settings. W. Thurston has used similar mappings in his study of diffeomorphisms of surfaces [9]. R. Easton has conjectured that linked twist maps are ergodic [4]. And a type of linked twist map occurs in the Stormer problem, the classical mechanical system describing the motion of a charged particle in the field of a magnetic dipole [3]. I. Linked twist mappings. Let A be an annulus in the plane centered at the origin and having outer radius 3 and inner radius p where 2 < p < 3. The twist map T of A is defined in polar coordinates by T(r, 0) = (r + T 3" ))pr() T fixes the boundaries of A and rotates each interior circle r = constant by an angle of 2v(r p)/(3 p). Also, T is an area-preserving diffeomorphism of A. Now let Ai, i = 1, 2, be two copies of A: A, centered at (1, 0), and A2 centered at (1, 0). The union A I U A2 is a space M which is homeomorphic to a closed two-disk minus three holes. Let Ti denote the twist map on Ai. Since Ti fixes the boundary of Ai, we may extend each Ti to all of M by defining Ti = identity on M Ai. The resulting maps are homeomorphisms of M, which we again denote by Ti. For each pair of integers j, k with j, k =# 0, we define a homeomorphism fjk = T ( I . We call fik a linked twist map. Presented to the Society, March 30, 1978; received by the editors August 12, 1977 and, in revised form, November 3, 1977. AMS (MOS) subject classifications (1970). Primary 58F15, 34C35; Secondary 54H20.

Subshifts of finite type in linked twist mappings