An Arc as the Inverse Limit of a Single Bonding Map of Type N on [0,1] |
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| Authors: | Email author" target="_blank">Hong?MoEmail author Song?Quan?Shi Fang?Ping?Zeng Jie?Hua?Mai |
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| Institution: | (1) Department of Applied Mathematics, Shanghai University of Finance Economics, Shanghai 200433, P. R. China;(2) The Key Laboratory of Complex System and Intelligent Science, Institute of Automation, Chinese Academy of Sciences, Beijing 100080, P. R. China;(3) The Key Laboratory of Complex System and Intelligent Science, Institute of Automation, Chinese Academy of Sciences, Beijing 100080, P. R. China;(4) Institute of Mathematics, Guangxi University, Nanning 530004, P. R. China;(5) Institute of Mathematics, Shantou University, Shantou 515062, P. R. China |
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| Abstract: | Let I = 0, 1], c
1, c
2 ∈ (0, 1) with c
1 < c
2 and f : I⊸I be a continuous map satisfying:
![$$
f\left| {_{{{\left {0,c_{1} } \right]}}} } \right.{\text{and}}f\left| {_{{{\left {c_{2} ,1} \right]}}} } \right.
$$](/content/481lb8tmtvjv9xv8/10114_2004_Article_363_TeX2GIFIEq1.gif)
are both strictly increasing and
![$$
f\left| {_{{{\left {c_{1} ,c_{2} } \right]}}} } \right.
$$](/content/481lb8tmtvjv9xv8/10114_2004_Article_363_TeX2GIFIEq2.gif)
is strictly decreasing. Let A = {x ∈ 0, c
1]∣f(x) = x},
a=max A, a
1 =max(A\{a}), and B = {x ∈ c
2, 1]∣f(x) = x}, b=minB, b
1 =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 c
1 < f (c
1) ≤ c
2 (resp. c
1 ≤ f (c
2) < c
2), 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 (c
2) > a (resp. f (c
1) < b).
(2) If f (c
1) ≤ c
1 (resp. f (c
2) ≥ c
2), then f has more than one fixed point, furthermore, if B≠φ and A\ {a} ≠φ, f (c
2) ≥ a or if a
1 < f (c
2) < a, f
2 (c
2) > f (c
2), (resp. f has more than one fixed point, furthermore, if A≠φ and B\{b}≠φ, f (c
1) ≤ b or if b < f (c
2) < b
1, f
2 (c
1) < f (c
1)).
(3) If f (c
1) > c
2 and f (c
2) < c
1, 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 ∈ c
1, c
2] such that f (u) = c
2 and f (v) = c
1.
Supported by the National Natural Science Foundation of China (No. 19961001, No. 60334020) and Outstanding Young Scientist
Research Fund. (No. 60125310) |
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| Keywords: | Inverse limit Arc Uninmodel map Periodic point |
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