Distributional chaos for operators with full scrambled sets

被引：33

作者：

Martinez-Gimenez, Felix ^{[1
]}

Oprocha, Piotr ^{[2
,3
]}

Peris, Alfredo ^{[1
]}

机构：

[1] Univ Politecn Valencia, Dept Matemat Aplicada, IUMPA, Valencia 46022, Spain

[2] AGH Univ Sci & Technol, Fac Appl Math, PL-30059 Krakow, Poland

[3] Polish Acad Sci, Inst Math, PL-00956 Warsaw, Poland

来源：

MATHEMATISCHE ZEITSCHRIFT | 2013年 / 274卷 / 1-2期

关键词：

Distributional chaos; Hypercyclic operators; Irregular vectors;

D O I：

10.1007/s00209-012-1087-8

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

In this article we answer in the negative the question of whether hypercyclicity is sufficient for distributional chaos for a continuous linear operator (we even prove that the mixing property does not suffice). Moreover, we show that an extremal situation is possible: There are (hypercyclic and non-hypercyclic) operators such that the whole space consists, except zero, of distributionally irregular vectors.

引用

页码：603 / 612

页数：10

共 19 条

[1] ON DEVANEY DEFINITION OF CHAOS [J].

BANKS, J ;

BROOKS, J ;

CAIRNS, G ;

DAVIS, G ;

STACEY, P .

AMERICAN MATHEMATICAL MONTHLY, 1992, 99 (04) :332-334

[2] Distributionally chaotic translation semigroups [J].

Barrachina, Xavier ;

Peris, Alfred .

JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS, 2012, 18 (04) :751-761

[3]

Bayart F, 2009, DYNAMICS LINEAR OPER

[4]

Beauzamy B., 1988, North-Holland Math Library

[5] Li-Yorke and distributionally chaotic operators [J].

Bermudez, T. ;

Bonilla, A. ;

Martinez-Gimenez, F. ;

Peris, A. .

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 2011, 373 (01) :83-93

[6] Topologically mixing hypercyclic operators [J].

Costakis, G ;

Sambarino, M .

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 2004, 132 (02) :385-389

[7]

Devaney R. L., 2018, An introduction to chaotic dynamical systems

[8] Hypercyclicity and supercyclicity for invertible bilateral weighted shift [J].

Feldman, NS .

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 2003, 131 (02) :479-485

[9]

Grosse-Erdmann K.-G., 2011, Linear Chaos

[10] Hypercyclic and chaotic weighted shifts [J].

Grosse-Erdmann, KG .

STUDIA MATHEMATICA, 2000, 139 (01) :47-68

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