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Let $f:X\to X$ be a continuous map on a compact metric space $(X, d)$ and for an arbitrary $x\in X$, \begin{equation*} \mathcal{SC}_d(x, f):=\{y\,|\, x \text{ can be strong $d$-chain to } y \}. \end{equation*} We give an example to show that $\mathcal{SC}_d(x, f)$ is dependent on the metric $d$ on $X$ but it is a closed and $f$-invariant set. We prove that if $\mathcal{SC}_d(x, f)\supseteq \Omega(f)$ or $f$ has the asymptotic-average shadowing property, then $\mathcal{SC}_d(x, f)= X$. Also, we show that if $f$ has the shadowing property, then $\limsup_{n\in \mathbb{N}}\{f^n\}= \mathcal{SC}_d(f)$ where $\mathcal{SC}_d(f)= \{(x, y) \,|\, y\in\mathcal{SC}_d(x, f)\}$. For each $n\in\mathbb{N}$, we give an example in which $\mathcal{SCR}_d(f^n)\neq \mathcal{SCR}_d(f).$ In spite of it, we prove that if $f^{-1}:(X, d)\to (X,d)$ is an equicontinuous map, then $\mathcal{SCR}_d(f^n)= \mathcal{SCR}_d(f)$ for all $n\in\mathbb{N}$.
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참고문헌 (11)
참고문헌 신청
R.M. Corless / 1995 / Approximate and Real Trajectories for Generic Dynamical Systems / Journal of Mathematical Analysis and Applications 189(2):409~423
R. Easton / 1977 / Chain transitivity and the domain of in uence of an invariant set, in The structure of attractors in dynamical systems (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), 95-102 / Lecture Notes in Math 668
Abbas Fakhari / 2010 / SOME PROPERTIES OF THE STRONG CHAIN RECURRENT SET / 대한수학회논문집 25(1):97~104
ALBERT FATHI / 2015 / Aubry–Mather theory for homeomorphisms / Ergodic Theory and Dynamical Systems 35(4):1187~1207
J. Franks / 1988 / A variation on the Poincare-Birkho theorem, in Hamiltonian dynamical sys-tems (Boulder, CO, 1987), 111-117 / Contemp. Math. 81
R. Easton / 1977 / Chain transitivity and the domain of in uence of an invariant set, in The structure of attractors in dynamical systems (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), 95-102 / Lecture Notes in Math 668
Abbas Fakhari / 2010 / SOME PROPERTIES OF THE STRONG CHAIN RECURRENT SET / 대한수학회논문집 25(1):97~104
ALBERT FATHI / 2015 / Aubry–Mather theory for homeomorphisms / Ergodic Theory and Dynamical Systems 35(4):1187~1207
J. Franks / 1988 / A variation on the Poincare-Birkho theorem, in Hamiltonian dynamical sys-tems (Boulder, CO, 1987), 111-117 / Contemp. Math. 81
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