A simple characterization of chaos for weighted composition 𝐶₀-semigroups on Lebesgue and Sobolev spaces

Kalmes, T.

doi:10.1090/proc/12794

A simple characterization of chaos for weighted composition $C_0$-semigroups on Lebesgue and Sobolev spaces

Author: T. Kalmes
Journal: Proc. Amer. Math. Soc. 144 (2016), 1561-1573
MSC (2010): Primary 47A16, 47D06; Secondary 35F15, 35F10
DOI: https://doi.org/10.1090/proc/12794
Published electronically: August 12, 2015
MathSciNet review: 3451233
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give a simple characterization of chaos for weighted composition $C_0$-semigroups on $L^p_\rho (\Omega )$ for an open interval $\Omega \subseteq \mathbb {R}$. Moreover, we characterize chaos for these classes of $C_0$-semigroups on the closed subspace $W^{1,p}_*(\Omega )$ of the Sobolev space $W^{1,p}(\Omega )$ for a bounded interval $\Omega \subset \mathbb {R}$. These characterizations simplify the characterization of chaos obtained by Aroza, Kalmes, and Mangino (2014) for these classes of $C_0$-semigroups.

References

Angela A. Albanese, Xavier Barrachina, Elisabetta M. Mangino, and Alfredo Peris, Distributional chaos for strongly continuous semigroups of operators, Commun. Pure Appl. Anal. 12 (2013), no. 5, 2069–2082. MR 3015670, DOI https://doi.org/10.3934/cpaa.2013.12.2069
Herbert Amann, Ordinary differential equations, De Gruyter Studies in Mathematics, vol. 13, Walter de Gruyter & Co., Berlin, 1990. An introduction to nonlinear analysis; Translated from the German by Gerhard Metzen. MR 1071170
Javier Aroza, Thomas Kalmes, and Elisabetta Mangino, Chaotic $C_0$-semigroups induced by semiflows in Lebesgue and Sobolev spaces, J. Math. Anal. Appl. 412 (2014), no. 1, 77–98. MR 3145782, DOI https://doi.org/10.1016/j.jmaa.2013.10.002
Javier Aroza and Alfred Peris, Chaotic behaviour of birth-and-death models with proliferation, J. Difference Equ. Appl. 18 (2012), no. 4, 647–655. MR 2905288, DOI https://doi.org/10.1080/10236198.2011.631535
Jacek Banasiak and Marcin Moszyński, Dynamics of birth-and-death processes with proliferation—stability and chaos, Discrete Contin. Dyn. Syst. 29 (2011), no. 1, 67–79. MR 2725281, DOI https://doi.org/10.3934/dcds.2011.29.67
Antoni Leon Dawidowicz and Anna Poskrobko, On chaotic and stable behaviour of the von Foerster-Lasota equation in some Orlicz spaces, Proc. Est. Acad. Sci. 57 (2008), no. 2, 61–69 (English, with English and Estonian summaries). MR 2554563, DOI https://doi.org/10.3176/proc.2008.2.01
R. deLaubenfels and H. Emamirad, Chaos for functions of discrete and continuous weighted shift operators, Ergodic Theory Dynam. Systems 21 (2001), no. 5, 1411–1427. MR 1855839, DOI https://doi.org/10.1017/S0143385701001675
Hassan Emamirad, Gisèle Ruiz Goldstein, and Jerome A. Goldstein, Chaotic solution for the Black-Scholes equation, Proc. Amer. Math. Soc. 140 (2012), no. 6, 2043–2052. MR 2888192, DOI https://doi.org/10.1090/S0002-9939-2011-11069-4
Karl-G. Grosse-Erdmann and Alfredo Peris Manguillot, Linear chaos, Universitext, Springer, London, 2011. MR 2919812
T. Kalmes, Hypercyclic, mixing, and chaotic $C_0$-semigroups induced by semiflows, Ergodic Theory Dynam. Systems 27 (2007), no. 5, 1599–1631. MR 2358980, DOI https://doi.org/10.1017/S0143385707000144
T. Kalmes, Hypercyclic $C_0$-semigroups and evolution families generated by first order differential operators, Proc. Amer. Math. Soc. 137 (2009), no. 11, 3833–3848. MR 2529893, DOI https://doi.org/10.1090/S0002-9939-09-09955-9
Elisabetta M. Mangino and Alfredo Peris, Frequently hypercyclic semigroups, Studia Math. 202 (2011), no. 3, 227–242. MR 2771652, DOI https://doi.org/10.4064/sm202-3-2
Mai Matsui, Mino Yamada, and Fukiko Takeo, Supercyclic and chaotic translation semigroups, Proc. Amer. Math. Soc. 131 (2003), no. 11, 3535–3546. MR 1991766, DOI https://doi.org/10.1090/S0002-9939-03-06960-0
Mai Matsui, Mino Yamada, and Fukiko Takeo, Erratum to: “Supercyclic and chaotic translation semigroups” [Proc. Amer. Math. Soc. 131 (2003), no. 11, 3535–3546; MR1991766], Proc. Amer. Math. Soc. 132 (2004), no. 12, 3751–3752. MR 2084100, DOI https://doi.org/10.1090/S0002-9939-04-07608-7
Ryszard Rudnicki, Chaoticity and invariant measures for a cell population model, J. Math. Anal. Appl. 393 (2012), no. 1, 151–165. MR 2921657, DOI https://doi.org/10.1016/j.jmaa.2012.03.055