Local Distributional Chaos

@article{65281,
   author = {Balibrea, Francisco and Rucká, Lenka},
   article_location = {Basel, Switzerland},
   article_number = {4},
   doi = {http://dx.doi.org/10.1007/s12346-022-00661-3},
   keywords = {Distributional chaos; Symbolic space; Chaos for interval maps; Chaos for triangular maps},
   language = {eng},
   issn = {1575-5460},
   journal = {Qualitative Theory of Dynamical Systems},
   title = {Local Distributional Chaos},
   url = {https://link.springer.com/article/10.1007/s12346-022-00661-3},
   volume = {21},
   year = {2022}
}

TY  - JOUR
ID  - 65281
AU  - Balibrea, Francisco - Rucká, Lenka
PY  - 2022
TI  - Local Distributional Chaos
JF  - Qualitative Theory of Dynamical Systems
VL  - 21
IS  - 4
SP  - "130-1"-"130-10"
EP  - "130-1"-"130-10"
PB  - Springer Basel AG
SN  - 15755460
KW  - Distributional chaos
KW  - Symbolic space
KW  - Chaos for interval maps
KW  - Chaos for triangular maps
UR  - https://link.springer.com/article/10.1007/s12346-022-00661-3
N2  - 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.
ER  - 

BALIBREA, Francisco a Lenka RUCKÁ. Local Distributional Chaos. \textit{Qualitative Theory of Dynamical Systems}. Basel, Switzerland: Springer Basel AG, 2022, roč.~21, č.~4, s.~''130-1''-''130-10'', 10 s. ISSN~1575-5460. Dostupné z: https://dx.doi.org/10.1007/s12346-022-00661-3.