Let I = [0, 1], c1, c2 ∈ (0, 1) with c1 < c2 and f : I⊸I be a continuous map satisfying:
are both strictly increasing and
is strictly decreasing. Let A = {x ∈ [0, c1]∣f(x) = x},
a=max A, a1 =max(A\{a}), and B = {x ∈ [c2, 1]∣f(x) = x}, b=minB, b1 =min(B\{b}). Then the inverse
limit (I, f) is an arc if and only if one of the following three conditions holds:
(1) If c1 < f (c1) ≤ c2 (resp. c1 ≤ f (c2) < c2), then f has a single fixed point, a period two orbit,
but no points of period greater than two or f has more than one fixed point but no points of other periods, furthermore, if A≠φ and B≠φ, then f (c2) > a (resp. f (c1) < b).
(2) If f (c1) ≤ c1 (resp. f (c2) ≥ c2), then f has more than one fixed point, furthermore, if B≠φ and A\ {a} ≠φ, f (c2) ≥ a or if a1 < f (c2) < a, f2 (c2) > f (c2), (resp. f has more than one fixed point, furthermore, if A≠φ and B\{b}≠φ, f (c1) ≤ b or if b < f (c2) < b1, f2 (c1) < f (c1)).
(3) If f (c1) > c2 and f (c2) < c1, then f has a single fixed point, a single period two orbit lying in I\(u, v) but no points of period greater than two, where u, v ∈ [c1, c2] such that f (u) = c2 and f (v) = c1.
Supported by the National Natural Science Foundation of China (No. 19961001, No. 60334020) and Outstanding Young Scientist
Research Fund. (No. 60125310) 相似文献
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