J 2022

Local Distributional Chaos

BALIBREA, Francisco and Lenka RUCKÁ

Basic information

Original name

Local Distributional Chaos

Authors

BALIBREA, Francisco (724 Spain) and Lenka RUCKÁ (203 Czech Republic, guarantor, belonging to the institution)

Edition

Qualitative Theory of Dynamical Systems, Basel, Switzerland, Springer Basel AG, 2022, 1575-5460

Other information

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10101 Pure mathematics

Country of publisher

Switzerland

Confidentiality degree

není předmětem státního či obchodního tajemství

RIV identification code

RIV/47813059:19610/22:A0000114

Organization unit

Mathematical Institute in Opava

UT WoS

000855766500001

Keywords in English

Distributional chaos; Symbolic space; Chaos for interval maps; Chaos for triangular maps

Tags

Tags

International impact, Reviewed
Změněno: 4/3/2023 12:19, Mgr. Aleš Ryšavý

Abstract

V originále

Distributional chaos was introduced in Schweizer and Smftal (Trans Am Math Soc 344:737-754, 1994) for continuous maps of the interval, as chaotic behavior based on development of distances between the orbits of points in the system. In Balibrea et al. (Chaos Solitons Fractals 23(5):1581-1583, 2005), this phenomenon was generalized to continuous maps of compact metric space and was distinguished into three different forms, chaos DC1, DC2 and DC3. In Loranty and Pawlak (Chaos 29:013104, 2019), the local idea of such behavior is studied, which leads to the definition of distributionally chaotic points (DC-points). It is also proved in Loranty and Pawlak (2019), that for interval maps, positive topological entropy implies existence of DC1-point. In this paper this result for interval maps is strengthened; it is proved that positive topological entropy implies existence of an uncountable set of DC1-points, and moreover this set can be chosen perfect. In greater dimensions than one, we deal with triangular systems on I-2 . In this case the relationship between topological entropy and different types of distributional chaos is not clearly understood and several different results are possible. In the paper we use an example of map F given by Kolyada (Ergod Theory Dyn Syst 12:749-768, 1992) to prove that the corresponding two dimensional system (I-2 , F) has positive topological entropy but without containing DC2-points, proving that there is no concentration of DC2-chaos.