By E. M. Friedlander, M. R. Stein
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One indirect means of dealing with this diﬃculty would be to employ a homeomorphism between P and Q and measure the distance between points of P by the distance between their images under the homeomorphism. Another solution in the case where all of the intervals [an , bn ] are nondegenerate, would be to use d(x, y) = i i>0 |xi − yi |/2 |ai − bi | as a metric for the inverse limit. Theorems 9, 10, and 15 also hold in this setting using more or less the same proofs and we use the corresponding more general results as needed without additional proof.
F | [a, c] is monotone 3. There is a positive integer j such that f j ([a, c]) = [a, b]. 26 1 Inverse Limits on Intervals Then, lim f is the closure of a ray having remainder a continuum K. ←− Moreover, K = lim k where k = f | [c, b]. ←− We have seen several theorems that give suﬃcient conditions for an inverse limit to be an indecomposable continuum. We end this section with a couple of theorems that furnish a suﬃcient condition for an inverse limit with a single bonding map to be decomposable.
Then f 2 (c1 ) ≤ b1 . Note that f 2 (p) = p and f 2 (b1 ) = p. In the previous paragraph we showed that (f 2 ) (x) < 0 for p ≤ x < c1 and (f 2 ) (x) < 0 for c1 < x ≤ b. It follows that f 2 ([p, b1 ]) ⊆ [p, b1 ]. Because S(f ) < 0, S(f 2 ) < 0. Thus, f 2 is strictly unimodal on [p, b1 ] with critical point c1 . With Devaney [11, p. 79], by a one-parameter family of functions gλ we mean a continuous function G deﬁned on a product of intervals such that (1) G(x, λ) = gλ (x), (2) the partial derivative of G in its second variable is continuous, and (3) gλ is a C ∞ function for each value of λ.