Abstract
Abstract First, we give the concepts of G-sequence shadowing property, G-equicontinuity and G-regularly recurrent point. Second, we study their dynamical properties in the inverse limit space under group action. The following results are obtained. (1) The self-mapping f f has the G-sequence shadowing property if and only if the shift mapping σ \sigma has the G ¯ \overline{G} -sequence shadowing property; (2) The self-mapping f f is G-equicontinuous if and only if the shift mapping σ \sigma is G ¯ \overline{G} -equicontinuous; (3) R R G ¯ ( σ ) = lim ← ( R R G ( f ) , f ) R{R}_{\overline{G}}\left(\sigma )=\underleftarrow{\mathrm{lim}}\left(R{R}_{G}(f),f) . These conclusions make up for the lack of theory in the inverse limit space under group action.
Highlights
The inverse limit space is a kind of very important space, which has always been the focus of research
Zhong and Wang [2] gave a sufficient and necessary condition for a point to be an equicontinuous point of dynamical system
In [3] it is shown that every ergodic invariant measure of a mean equicontinuous system has discrete spectrum; Ji, Chen and Zhang [4] proved that the shift map has the Lipschitz shadowing property if and only if the self-map has the Lipschitz shadowing property in the inverse limit space
Summary
The inverse limit space is a kind of very important space, which has always been the focus of research. Scholar put forward the concept of the inverse limit space under group action and proved that the shift mapping and the self-mapping are equivariant to each other in G-shadowing property and G-strong shadowing property, see [1]. In [3] it is shown that every ergodic invariant measure of a mean equicontinuous system has discrete spectrum; Ji, Chen and Zhang [4] proved that the shift map has the Lipschitz shadowing property if and only if the self-map has the Lipschitz shadowing property in the inverse limit space. First, we give the concepts of G-sequence shadowing property, G-equicontinuity and G-regularly recurrent point We study their dynamical properties in the inverse limit space under group action and will get the following theorem.
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