1. ADJUSTE D M A P S In this section we set up an approach to determining forcing relations based on ideas in [ALM] (where only cycles were considered). Suppose / G £(I) exhibits the pattern 6 G ^Pn on the finite set V {x\ X2 xn}. By a "P-interval we mean any of the closed intervals [xi, #,-+1] bounded by adjacent elements of V. The map / is TMinear (resp P - m o n o t o n e ) if V includes the endpoints of 7 and / is afflne (resp. monotone) on each V- interval. When we wish to specify the pattern 6 exhibited by / o n ? , we will call / ^-linear on P(resp. 0-monoton e on V). Note that any two ^-linear maps are topologically conjugate (see [BlCvl] for a proof) and in particular they are all conjugate to the canonical 0-linear m a p for which V = {1, . . . , n } . (The analogous statement for #-monotone maps is clearly false.) Given / G £(7), a (nondegenerate) interval [01,02] C 7 /-cover s another (nondegenerate) interval [61,62] with inde x a = ± 1 if f(a\) a 6 a /(02) for i = 1,2 (see the introduction for the notation ?). A closed interval 7 mini- mally /-cover s the interval J if 7 /-covers J and no proper closed subinterval of 7 /-covers J. Note that this is equivalent to the conditions that (i) no interior point of 7 maps to the boundary of J and (a) /(/) = J. The following observations will play a central role in our arguments. 1.1. LEMMA. Suppose f G £ and I [01,02] f-covers J = [61,62] with index a - ± 1 . Then: (i) given any finite subset yi y2 ' ' - Vk of J there exists a finite subset x\ o %2 7 o Xk of I with f(xj) = yj for j = 1,. . ., k (ii) there exists a subinterval 7 C 7 which minimally f-covers J with index J. P R O O F : Proof of (i): We use induction on k. The existence of x\ follows from the intermediate-value theorem. Suppose we have x\ a .. . a Xk-i with f(xi) yi. Let a be the endpoint of 7 with Xk-i a a , and 7^ the interval with endpoints Xk-ij oc. Then 7^ /-covers [y^_i,62] with index 0*, hence contains a pre-image Xk of yk] since x^-i & Xk ? ct, we are done. Proof of (ii): Since f(a\) a b\ a /(02), the compact set / - 1 [6i ] is nonempty let a be its cr -maximal element. The set {x G 7 | a a x and f(x) 62} is compact and contains an endpoint of 7 let f3 be its cr-minimal element. Then a a /?, no element a a x a (3 can map to either endpoint of J, and a, (3 map to the endpoints of J. Thus the interval 7 with endpoints a, (3 /-covers J minimally with index a. | Now, given / G £(7), exhibiting 0 G tyn on V {x\ x2 xn}, we can keep track of /-orbits in I-p [xi,xn] via itineraries. Abstractly, an itinerar y
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