Chaotic sets of continuous and discontinuous maps

被引：2

作者：

Fu, Xin-Chu ^{[1
]}

Chen, Zhan-He ^{[1
]}

Gao, Hongjun ^{[2
]}

Li, Chang-Pin ^{[1
]}

Liu, Zeng-Rong ^{[1
]}

机构：

[1] Shanghai Univ, Dept Math, Shanghai 200444, Peoples R China

[2] Nanjing Normal Univ, Dept Math, Nanjing 210097, Peoples R China

来源：

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS | 2010年 / 72卷 / 01期

关键词：

Chaos; Scrambled set; Shift; Continuous map; Discontinuous map; Piecewise isometry; PLANAR PIECEWISE ISOMETRIES; SCRAMBLED SETS; DIGITAL-FILTERS; TOPOLOGICAL-ENTROPY; DYNAMICAL-SYSTEMS; SYMBOLIC ANALYSIS; LEBESGUE MEASURE; OMEGA-CHAOS; EVERYWHERE; SUBSHIFTS;

D O I：

10.1016/j.na.2009.06.075

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

This paper discusses Li-Yorke chaotic sets of continuous and discontinuous maps with particular emphasis to shift and subshift maps. Scrambled sets and maximal scrambled sets are introduced to characterize Li-Yorke chaotic sets. The orbit invariant for a scrambled set is discussed. Some properties about maximality, equivalence and uniqueness of maximal scrambled sets are also discussed. It is shown that for shift maps the set of all scrambled pairs has full measure and chaotic sets of some discontinuous maps, such as the Gauss map, interval exchange transformations, and a class of planar piecewise isometries, are studied. Finally, some open problems on scrambled sets are listed and remarked. (C) 2009 Elsevier Ltd. All rights reserved.

引用

页码：399 / 408

页数：10

共 50 条

[31] The construction of chaotic maps in the sense of Devaney on dendrites which commute to continuous maps on the unit interval
Arai, T
Chinen, N
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 2004, 11 (2-3) : 547 - 556
[32] Chaotic invariant sets of high-dimensional Henon-like maps
Qin, WX
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 2001, 264 (01) : 76 - 84
[33] The Topological Entropy of Cyclic Permutation Maps and Some Chaotic Properties on Their MPE sets
Li, Risong
Lu, Tianxiu
COMPLEXITY, 2020, 2020
[34] Quantum Image Cryptography of Continuous Chaotic Maps by Using Pixel Shuffeling
Satre, Shilpa M.
Joshi, Bharti
JOURNAL OF ELECTRICAL SYSTEMS, 2024, 20 (05) : 463 - 476
[35] On the Complete Chaotic Maps that Preserve Prescribed Absolutely Continuous Invariant Densities
Huang, W.
TOPICS ON CHAOTIC SYSTEMS, 2009, : 166 - 173
[36] Criteria for a general continuous self-map to have chaotic sets
Fu, Xin-Chu
Nonlinear Analysis, Theory, Methods and Applications, 1996, 26 (02): : 329 - 334
[37] Criteria for a general continuous self-map to have chaotic sets
Fu, XC
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 1996, 26 (02) : 329 - 334
[38] CONTINUOUS CHAOTIC FUNCTIONS OF AN INTERVAL HAVE GENERICALLY SMALL SCRAMBLED SETS
MIZERA, I
BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY, 1988, 37 (01) : 89 - 92
[39] Behavior of Dynamical Systems Subjected to Continuous and Discontinuous Forcing: Application to Laminar Chaotic Mixing
Perez, A. T.
Chacon, R.
Castellanos, A.
International Journal of Bifurcations and Chaos in Applied Sciences and Engineering, 1996, 6 (12-B):
[40] Continuous maps of dendrites with finitely many branch points and nonwandering sets
Arai, T
Chinen, N
Suda, T
TOPOLOGY PROCEEDINGS, VOL 26, NO 1, 2001-2002, 2002, 26 (01): : 29 - 36

← 1 2 3 4 5 →