Distributional chaos for backward shifts

AbstractWe provide sufficient conditions which give uniform distributional chaos for backward shift operators. We also compare distributional chaos with other well-studied notions of chaos for linear operators, like Devaney chaos and hypercyclicity, and show that Devaney chaos implies uniform distributional chaos for weighted backward shifts, but there are examples of backward shifts which are uniformly distributionally chaotic and not hypercyclic

The Specification Property and Infinite Entropy for Certain Classes of Linear Operators

We study the specification property and infinite topological entropy for two specific types of linear operators: translation operators on weighted Lebesgue function spaces and weighted backward shift operators on sequence F-spaces. It is known from the work of Bartoll, Martinínez-Giménez, Murillo-Arcila (2014), and Peris, that for weighted backward shift operators, the existence of a single non-trivial periodic point is sufficient for specification. We show this also holds for translation operators on weighted Lebesgue function spaces. This implies, in particular, that for these operators, the specification property is equivalent to Devaney chaos. We also show that these forms of chaos imply infinite topological entropy, but that the converse does not hold

Chaotic differential operators

We give sufficient conditions for chaos of (differential) operators on Hilbert spaces of entire functions. To this aim we establish conditions on the coefficients of a polynomial P(z) such that P(B) is chaotic on the space lp, where B is the backward shift operator. © 2011 Springer-Verlag.This work was partially supported by the MEC and FEDER Projects MTM2007-64222, MTM2010-14909, and by GVA Project GV/2010/091, and by UPV Project PAID-06-09-2932. The authors would like to thank A. Peris for helpful comments and ideas that produced a great improvement of the paper's presentation. We also thank the referees for their helpful comments and for reporting to us a gap in Theorem 1.Conejero Casares, JA.; Martínez Jiménez, F. (2011). Chaotic differential operators. Revista- Real Academia de Ciencias Exactas Fisicas Y Naturales Serie a Matematicas. 105(2):423-431. https://doi.org/10.1007/s13398-011-0026-6S4234311052Bayart, F., Matheron, É.: Dynamics of Linear Operators, Cambridge Tracts in Mathematics, vol. 179. Cambridge University Press, Cambridge (2009)Bermúdez T., Miller V.G.: On operators T such that f(T) is hypercyclic. Integr. Equ. Oper. Theory 37(3), 332–340 (2000)Bonet J., Martínez-Giménez F., Peris A.: Linear chaos on Fréchet spaces. Int. J. Bifur. Chaos Appl. Sci. Eng. 13(7), 1649–1655 (2003)Chan K.C., Shapiro J.H.: The cyclic behavior of translation operators on Hilbert spaces of entire functions. Indiana Univ. Math. J. 40(4), 1421–1449 (1991)Conejero J.A., Müller V.: On the universality of multipliers on $${\mathcal{H}({\mathbb {C}})}$$ . J. Approx. Theory. 162(5), 1025–1032 (2010)deLaubenfels R., Emamirad H.: Chaos for functions of discrete and continuous weighted shift operators. Ergodic Theory Dyn. Syst. 21(5), 1411–1427 (2001)Devaney, R.L.: An introduction to chaotic dynamical systems, 2nd edn. In: Addison-Wesley Studies in Nonlinearity. Addison-Wesley Publishing Company Advanced Book Program, Redwood City (1989)Godefroy G., Shapiro J.H.: Operators with dense, invariant, cyclic vector manifolds. J. Funct. Anal. 98(2), 229–269 (1991)Grosse-Erdmann K.-G.: Hypercyclic and chaotic weighted shifts. Stud. Math. 139(1), 47–68 (2000)Grosse-Erdmann, K.-G., Peris, A.,: Linear chaos. Universitext, Springer, New York (to appear, 2011)Herzog G., Schmoeger C.: On operators T such that f(T) is hypercyclic. Stud. Math. 108(3), 209–216 (1994)Kahane, J.-P.: Some random series of functions, 2nd edn. In: Cambridge Studies in Advanced Mathematics, vol. 5. Cambridge University Press, Cambridge (1985)Martínez-Giménez F., Peris A.: Chaos for backward shift operators. Int. J. Bifur. Chaos Appl. Sci. Eng. 12(8), 1703–1715 (2002)Martínez-Giménez F.: Chaos for power series of backward shift operators. Proc. Am. Math. Soc. 135, 1741–1752 (2007)Müller V.: On the Salas theorem and hypercyclicity of f(T). Integr. Equ. Oper. Theory 67(3), 439–448 (2010)Protopopescu V., Azmy Y.Y.: Topological chaos for a class of linear models. Math. Models Methods Appl. Sci. 2(1), 79–90 (1992)Rolewicz S.: On orbits of elements. Stud. Math. 32, 17–22 (1969)Salas H.N.: Hypercyclic weighted shifts. Trans. Am. Math. Soc. 347(3), 93–1004 (1995)Shapiro, J.H.: Simple connectivity and linear chaos. Rend. Circ. Mat. Palermo. (2) Suppl. 56, 27–48 (1998

Shift-like Operators on Lp(X)L^p(X)

In this article we develop a general technique which takes a known characterization of a property for weighted backward shifts and lifts it up to a characterization of that property for a large class of operators on $L^p(X)$. We call these operators ``shift-like''. The properties of interest include chaotic properties such as Li-Yorke chaos, hypercyclicity, frequent hypercyclicity as well as properties related to hyperbolic dynamics such as shadowing, expansivity and generalized hyperbolicity. Shift-like operators appear naturally as composition operators on $L^p(X)$ when the underlying space is a dissipative measure system. In the process of proving the main theorem, we provide some results concerning when a property is shared by a linear dynamical system and its factors.Comment: arXiv admin note: text overlap with arXiv:2009.1152

The specification property for backward shifts

This is an Accepted Manuscript of an article published by Taylor & Francis Group in [Journal of Difference Equations and Applications] on [2012], available online at: http://www.tandfonline.com/10.1080/10236198.2011.586636We characterize when backward shift operators defined on Banach sequence spaces exhibit the strong specification property. In particular, within this framework, the specification property is equivalent to the notion of chaos introduced by Devaney.This work was supported in part by MEC and FEDER, Project MTM2010-14909, and by Generalitat Valenciana, Project PROMETEO/2008/101.Bartoll Arnau, S.; Martínez Jiménez, F.; Peris Manguillot, A. (2012). The specification property for backward shifts. Journal of Difference Equations and Applications. 18(4):599-605. https://doi.org/10.1080/10236198.2011.586636S599605184Bauer, W., & Sigmund, K. (1975). Topological dynamics of transformations induced on the space of probability measures. Monatshefte für Mathematik, 79(2), 81-92. doi:10.1007/bf01585664Bermúdez, T., Bonilla, A., Conejero, J. A., & Peris, A. (2005). Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces. Studia Mathematica, 170(1), 57-75. doi:10.4064/sm170-1-3Bonet, J., Martínez-Giménez, F., & Peris, A. (2001). A Banach Space which Admits No Chaotic Operator. Bulletin of the London Mathematical Society, 33(2), 196-198. doi:10.1112/blms/33.2.196Chan, K., & Shapiro, J. (1991). Indiana University Mathematics Journal, 40(4), 1421. doi:10.1512/iumj.1991.40.40064Costakis, G., & Sambarino, M. (2004). Proceedings of the American Mathematical Society, 132(02), 385-390. doi:10.1090/s0002-9939-03-07016-3Denker, M., Grillenberger, C., & Sigmund, K. (1976). Ergodic Theory on Compact Spaces. Lecture Notes in Mathematics. doi:10.1007/bfb0082364Godefroy, G., & Shapiro, J. H. (1991). Operators with dense, invariant, cyclic vector manifolds. Journal of Functional Analysis, 98(2), 229-269. doi:10.1016/0022-1236(91)90078-jGrosse-Erdmann, K.-G. (2000). Hypercyclic and chaotic weighted shifts. Studia Mathematica, 139(1), 47-68. doi:10.4064/sm-139-1-47-68Grosse-Erdmann, K.-G., & Peris, A. (2010). Weakly mixing operators on topological vector spaces. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 104(2), 413-426. doi:10.5052/racsam.2010.25Grosse-Erdmann, K.-G., & Peris Manguillot, A. (2011). Linear Chaos. Universitext. doi:10.1007/978-1-4471-2170-1Lampart, M., & Oprocha, P. (2009). Shift spaces, ω-chaos and specification property. Topology and its Applications, 156(18), 2979-2985. doi:10.1016/j.topol.2009.04.063MARTÍNEZ-GIMÉNEZ, F., & PERIS, A. (2002). CHAOS FOR BACKWARD SHIFT OPERATORS. International Journal of Bifurcation and Chaos, 12(08), 1703-1715. doi:10.1142/s0218127402005418Oprocha, P. (2007). Specification properties and dense distributional chaos. Discrete and Continuous Dynamical Systems, 17(4), 821-833. doi:10.3934/dcds.2007.17.821Oprocha, P., & Štefánková, M. (2008). Specification property and distributional chaos almost everywhere. Proceedings of the American Mathematical Society, 136(11), 3931-3940. doi:10.1090/s0002-9939-08-09602-0Peris, A., & Saldivia, L. (2005). Syndetically Hypercyclic Operators. Integral Equations and Operator Theory, 51(2), 275-281. doi:10.1007/s00020-003-1253-9Sigmund, K. (1974). On dynamical systems with the specification property. Transactions of the American Mathematical Society, 190, 285-285. doi:10.1090/s0002-9947-1974-0352411-

Dynamics of shift operators on non-metrizable sequence spaces

[EN] We investigate dynamical properties such as topological transitivity, (sequential) hypercyclicity, and chaos for backward shift operators associated to a Schauder basis on LF-spaces. As an application, we characterize these dynamical properties for weighted generalized backward shifts on Kothe coechelon sequence spaces k(p)((v((m)))(m is an element of N)) in terms of the defining sequence of weights (v((m)))(m) (is an element of N). We further discuss several examples and show that the annihilation operator from quantum mechanics is mixing, sequentially hypercyclic, chaotic, and topologically ergodic on S'(R).The first author was partially supported by MICINN and FEDER, Proj. MTM2016-76647-P, and by Generalitat Valenciana, Project PROMETEO/2017/102. The research of the third author was partially supported by MICINN and FEDER, Projects MTM2016-75963-P and PID2019105011GB-I00, and by Generalitat Valenciana, Project PROMETEO/2017/102.Bonet Solves, JA.; Kalmes, T.; Peris Manguillot, A. (2021). Dynamics of shift operators on non-metrizable sequence spaces. Revista Matemática Iberoamericana. 37(6):2373-2397. https://doi.org/10.4171/rmi/12672373239737