Hypercyclic operators for iterated function systems

Temel Bulgular & Veri:We prove that for a linear IFS, hypercyclicity implies sensitivity and if an IFS is abelian, then hypercyclicity also implies multi-sensitivity and hence thick sensitivity.

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Sonuç & Katkı:We also give some equivalent conditions for hypercyclicity as well as weakly mixing for an IFS of operators.

🔬Tür:Araştırma Makalesi

📍Kapsam:Genişletilmiş Akademik Örneklem

⚙️Değişkenler:Iterated function systems · hypercyclic operators · sensitive operators

Abstract

In this paper we introduce and study the notion of hypercyclicity for iterated function systems (IFS) of operators. We prove that for a linear IFS, hypercyclicity implies sensitivity and if an IFS is abelian, then hypercyclicity also implies multi-sensitivity and hence thick sensitivity. We also give some equivalent conditions for hypercyclicity as well as weakly mixing for an IFS of operators.

Anahtar kelimeler

Iterated function systems hypercyclic operators sensitive operators weakly mixing operators

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Bu Makaleye Atıf Yap

Salman, M., Das, R. (2022). Hypercyclic operators for iterated function systems. Hacettepe Journal of Mathematics and Statistics, Cilt: 50 - Sayı: 2, 483–491. https://ajindex.com/articles/529548

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Kaynakça (20)

A.Z. Bahabadi, Shadowing and average shadowing properties for iterated function systems, Georgian Math. J. 22, 179–184, 2015
A.Z. Bahabadi, On chaos for iterated function systems, Asian-Eur. J. Math. 11, 1850054, 2018
F. Bayart and É. Matheron, Dynamics of linear operators, 179, Cambridge University Press, Cambridge, 2009
G. Costakis and A. Manoussos, J-class operators and hypercyclicity, J. Operator Theory, 67, 101–119, 2012
J.H. Elton and M. Piccioni, Iterated function systems arising from recursive estima- tion problems, Probab. Theory Related Fields, 91, 103–114, 1992
B. Forte and E.R. Vrscay, Solving the inverse problem for function/image approxi- mation using iterated function systems. II. Algorithm and computations, Fractals, 2, 335–346, 1994
F.H. Ghane, E. Rezaali, and A. Sarizadeh, Sensitivity of iterated function systems, J. Math. Anal. Appl. 469, 493–503, 2019
K-G. Grosse-Erdmann and A. Peris, Weakly mixing operators on topological vector spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 104, 413–426, 2010
K-G. Grosse-Erdmann and A. Peris, Linear chaos, Springer Science & Business Media, 2011
J.E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30, 713–747, 1981
M. Kostić, Chaos for linear operators and abstract differential equations, Nova Science Publishers, New York, 2020
C. Ma and P. Zhu, A remark on sensitivity and Li-Yorke sensitivity of iterated func- tion systems, Qual. Theory Dyn. Syst. 18, 1–9, 2019
M. Mohtashamipour and A.Z. Bahabadi, Accessibility on iterated function systems, Georgian Math. J. 28 (1), 117–124, 2021
T.K.S. Moothathu, Stronger forms of sensitivity for dynamical systems, Nonlinearity, 20, 2115–2126, 2007
M. Murillo-Arcila and A. Peris, Chaotic behaviour on invariant sets of linear opera- tors, Integral Equations Operator Theory, 81, 483–497, 2015
M. Salman and R. Das, Furstenberg family and multi-sensitivity in non-autonomous systems, J. Difference Equ. Appl. 25, 1755–1767, 2019
M. Salman and R. Das, Multi-transitivity in non-autonomous discrete systems, Topol- ogy Appl. 278, 107237, 2020
M. Salman and R. Das, Sensitivity and property $P$ in non-autonomous systems, Mediterr. J. Math. 17, 128, 2020
X.Wu, S. Liang, Y. Luo, M. Xin and X. Zhang, A remark on limit shadowing property for iterated function systems, U.P.B. Sci. Bull. Series A, Appl. Math. Phys. 81, 107– 114, 2019
X. Wu