Abstract
Let G be a graph and f: G → G be a continuous map. Denote by h(f), P(f),AP(f),R(f) and ω(x, f) the topological entropy of f, the set of periodic points of f, the set of almost periodic points of f, the set of recurrent points of f and the ω-limit set of x under f, respectively. In this paper, we show that the following statements are equivalent: (1) h(f) > 0. (2) There exists an x ∈ G such that ω(x, f) ∩ P(f) ≠ Ø and ω(x, f) is an infinite set. (3) There exists an x ∈ G such that ω(x, f) contains two minimal sets. (4) There exist x, y ∈ G such that ω(x, f) − ω(y, f) is an uncountable set and ω(y, f) ∩ ω(x, f) ≠ Ø. (5) There exist an x ∈ G and a closed subset A ⊂ ω(x, f) with f(A) ⊂ A such that ω(x, f) − A is an uncountable set. (6) R(f) − AP(f) ≠ Ø. (7) f| P ( f ) is not pointwise equicontinuous.
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We thank the referees for their careful reading of the manuscript and constructive comments and suggestions.
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Supported by NNSF of China (Grant No. 11761011) and NSF of Guangxi (Grant Nos. 2016GXNSFBA380235 and 2016GXNSFAA380286) and YMTBAPP of Guangxi Colleges (Grant No. 2017KY0598) and SF of Guangxi University of Finance and Economics (Grant No. 2017QNA04)
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Sun, T.X. Topological entropy of a graph map. Acta. Math. Sin.-English Ser. 34, 194–208 (2018). https://doi.org/10.1007/s10114-017-7236-6
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DOI: https://doi.org/10.1007/s10114-017-7236-6