Detailed Information on Publication Record
2022
Local Distributional Chaos
BALIBREA, Francisco and Lenka RUCKÁBasic 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
References:
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.