The Research of Sequence Shadowing Property and Regularly Recurrent Point on the Double Inverse Limit Space

Abstract

According to the definition of sequence shadowing property and regularly recurrent point in the inverse limit space, we introduce the concept of sequence shadowing property and regularly recurrent point in the double inverse limit space and study their dynamical properties. The following results are obtained: (1) Regularly recurrent point sets of the double shift map σ f ∘ σ g are equal to the double inverse limit space of the double self-map f ∘ g in the regularly recurrent point sets. (2) The double self-map f ∘ g has sequence shadowing property if and only if the double shift map σ f ∘ σ g has sequence shadowing property. Thus, the conclusions of sequence shadowing property and regularly recurrent point are generalized to the double inverse limit space.

Dynamical Property of the Shift Map under Group Action

“…At present, shadowing property has gradually become an important theory and concept in dynamical system. Relevant results are seen in [1][2][3][4][5][6][7][8][9][10][11]. For example, Wang and Zeng [1] proved that if f has q-average tracking property, then f is chain transitivity under some conditions.…”

“…Wu [2] showed that f has tracking property and σ has tracking property are equivalent. Ji et al [3] proved that f ∘ g has the Lipschitz shadowing property and σ f ∘g has the Lipschitz shadowing property are equivalent in the double inverse limit space. Ahmadi and Hosseini [4] gave that δ-ergodic pseudo-orbit of a system means chain mixing.…”

“…The data used to support the findings of this study are included within references [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] in the article.…”