Equicontinuity of maps on a dendrite with finite branch points |
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Authors: | Tai Xiang Sun Guang Wang Su Hong Jian Xi Xin Kong |
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Institution: | 1. College of Information and Statistics, Guangxi University of Finance and Economics, Nanning 530003, P. R. China;2. College of Mathematics and Information Science, Guangxi University, Nanning 530004, P. R. China |
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Abstract: | Let (T, d) be a dendrite with finite branch points and f be a continuous map from T to T. Denote by ω(x, f) and P(f) the ω-limit set of x under f and the set of periodic points of f, respectively. Write Ω(x, f) = {y| there exist a sequence of points x k ∈ T and a sequence of positive integers n 1 < n 2 < ··· such that lim k→∞ x k = x and lim k→∞ \(f^{n_{k}}\) (x k ) = y}. In this paper, we show that the following statements are equivalent: (1) f is equicontinuous. (2) ω(x, f) = Ω(x, f) for any x ∈ T. (3) ∩ n=1 ∞ f n (T) = P(f), and ω(x, f) is a periodic orbit for every x ∈ T and map h: x → ω(x, f) (x ∈ T) is continuous. (4) Ω(x, f) is a periodic orbit for any x ∈ T. |
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Keywords: | Dendrite map equicontinuity periodic point ω-limit set |
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