Topological Ergodic Shadowing and Chaos on Uniform Spaces

Abstract

This paper firstly proves that every dynamical system defined on a Hausdorff uniform space with topologically ergodic shadowing is topologically mixing, thus topologically chain mixing. Then, the following is proved: (1) every weakly mixing dynamical system defined on a second countable Baire–Hausdorff uniform space is chaotic in the sense of both Li–Yorke and Auslander–Yorke; (2) every point transitive dynamical system defined on a Hausdorff uniform space is either almost equicontinuous or sensitive.

Some Properties of Recurrent Sets for Endomorphisms of Topological Groups

“…In the presence of compactness, existence of an unique uniformity, allows ue to mimic existing metric proofs. The uniform approach has been studied in a number of cases: Hood [15] defined topological entropy for uniform spaces; Morales and Sirvent [16] considered positively expansive measures for measurable functions on uniform spaces, extending results from the literature; Devaney chaos for uniform spaces is considered in [9]; Auslander et al [3] generalized many known results about equicontinuity to the uniform spaces; Das et al [10] generalized spectral decomposition theorem to the uniform spaces; We [1] generalized concepts of entropy points, expansivity and shadowing property for dynamical systems on uniform spaces and we obtained a relation between topological shadowing property and positive uniform entropy;; Wu et al [22] obtained that every point transitive dynamical system defined on a Hausdorff uniform space is either almost equicontinuous or sensitive.…”

Recurrent sets and shadowing for finitely generated semigroup actions on metric spaces

“…A dynamical system has the shadowing property if every sufficiently precise trajectory is closed to some exact trajectory. The shadowing property has been developed intensively in recent years, and many authors obtained results about chaos and stability by studying the various type of shadowing (see [1,11,17,19,20,22,24,25,[27][28][29]). Wu et al [25] introduced the notion of M α -shadowing and proved that a dynamical system has the average shadowing property if and only if it has the M α -shadowing property for any α ∈ [0, 1).…”

Shifts, rotations and distributional chaos

“…The notion of Li-Yorke sensitivity was firstly introduced by Akin and Kolyada [1] in 2003. More recent results on chaos can be found in [2,3,7,17,18,[20][21][22][23][24][25][26][27][28][29][30][31].…”