0. INTRODUCTION The celebrated theorem of Sharkovskii [Sa] illustrates the rigid restrictions on the set of periodic orbits of a continuous interval map that are imposed by the ordering of points along a line. While the statement of Sharkovskii's theorem focuses on the least periods of these orbits, its various proofs [Sa St Str HM Bu B G M Y ] indicate a rich combinatorial structure controlling the disposition of the orbits themselves. Various features of this structure have been elucidated by many workers, beginning with Sharkovskii (and, in a more limited context, Myrberg [My] and Metropolis, Stein and Stein [MSS]), although the general problem of understanding this structure was explicitly formulated only recently by Baldwin [Ba]. Baldwin's work, as well as other recent papers in this area [ALM ALS B e l - 3 BIC BIH C Ca J FO-2] have focused on the information inherent in a single periodic orbit. In the present paper, we broaden the scope of this study to encompass arbitrary finite, invariant sets. This leads us to study a relation we call forcing on abstract combinatorial objects we call combinatorial patterns. The combinatorial structure we wish to study can be set up as follows. Given / : / —• I a continuous map of a closed interval to itself and V a finite /-invariant set (i.e., f[V] CV), label the elements of V Pi P2 " Pn- Then the action of / on V can be codified in the map 0 : { l , . . . , n } - + { l , . . . , n } defined by f(Pi) =Pe(i) i = l , . . . , n . The finiteness of V insures that it consists of finitely many preperiodic /-orbits the map 9 encodes the combinatorial structure of each orbit and the way these orbits intertwine. To stress the combinatorial role of 9, we refer to any map of { 1 , . . . , 7i} to itself as a combinatorial pattern on n elements, or a pattern for short. The number n is the degree of 9. We say that the map / exhibits the combinatorial pattern 9 on V, and call V a representative of 9 in / . A given finite invariant set V represents a unique combinatorial pattern 9, but a given combinatorial pattern 9 may have many representatives in / . When V is a single periodic orbit, the combinatorial pattern 9 represented by V is a cyclic permutation. More generally, a bijective combinatorial pattern 9 is a (possibly multicyclic) permutation, and a representative of 9 is a finite union of periodic orbits. Received by editor February 1, 1990. Received in revised form November 15, 1990.
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