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 57 条

[41] Continuous maps with the whole space being a scrambled set [J].

Mai, JH .

CHINESE SCIENCE BULLETIN, 1997, 42 (19) :1603-1606

[42]

MISIUREWICZ M, 1985, LECT NOTES MATH, V1163, P125

[43] CONTINUOUS CHAOTIC FUNCTIONS OF AN INTERVAL HAVE GENERICALLY SMALL SCRAMBLED SETS [J].

MIZERA, I .

BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY, 1988, 37 (01) :89-92

[44]

MIZERA I, 1989, DYNAMICAL SYSTEMS ER, P161

[45] COMPLEX BEHAVIOR IN DIGITAL FILTERS [J].

Ogorzalek, Maciej J. .

INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, 1992, 2 (01) :11-29

[46] MEASURES OF CHAOS AND A SPECTRAL DECOMPOSITION OF DYNAMICAL-SYSTEMS ON THE INTERVAL [J].

SCHWEIZER, B ;

SMITAL, J .

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 1994, 344 (02) :737-754

[47] Hamiltonian mappings and circle packing phase spaces [J].

Scott, AJ ;

Holmes, CA ;

Milburn, GJ .

PHYSICA D, 2001, 155 (1-2) :34-50

[48] A CHAOTIC FUNCTION WITH SOME EXTREMAL PROPERTIES [J].

SMITAL, J .

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 1983, 87 (01) :54-56

[49] CHAOTIC FUNCTIONS WITH ZERO TOPOLOGICAL-ENTROPY [J].

SMITAL, J .

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 1986, 297 (01) :269-282

[50]

Smítal J, 2003, DISCRETE CONT DYN S, V9, P1323

← 1 2 3 4 5 6 →