Universality and chaos for tensor products of operators

AbstractWe give sufficient conditions for the universality of tensor products {Tn⊗Rn:n∈N} of sequences of operators defined on Fréchet spaces. In particular we study when the tensor product T⊗R of two operators is chaotic in the sense of Devaney. Applications are given for natural operators on function spaces of several variables, in Infinite Holomorphy, and for multiplication operators on the algebra L(E) following the study of Kit Chan

Mixing properties for nonautonomous linear dynamics and invariant sets

We study mixing properties (topological mixing and weak mixing of arbitrary order) for nonautonomous linear dynamical systems that are induced by the corresponding dynamics on certain invariant sets. The kinds of nonautonomous systems considered here can be defined using a sequence of linear operators on a topological vector space X such that there is an invariant set Y for which the dynamics restricted to Y satisfies a certain mixing property. We then obtain the corresponding mixing property on the closed linear span of Y. We also prove that the class of nonautonomous linear dynamical systems that are weakly mixing of order nn contains strictly the corresponding class with the weak mixing property of order n+1.This work was supported in part by MEC and FEDER, Project MTM2010-14909, and by GV, Project PROMETEO/2008/101. The first author was also supported by a grant from the FPU Program of MEC. We thank the referees whose reports led to an improvement in the presentation of this work.Murillo Arcila, M.; Peris Manguillot, A. (2013). Mixing properties for nonautonomous linear dynamics and invariant sets. Applied Mathematics Letters. 26(2):215-218. https://doi.org/10.1016/j.aml.2012.08.014S21521826

On the existence of polynomials with chaotic behaviour

We establish a general result on the existence of hypercyclic (resp., transitive, weakly mixing, mixing, frequently hypercyclic) polynomials on locally convex spaces. As a consequence we prove that every (real or complex) infinite-dimensional separable Frèchet space admits mixing (hence hypercyclic) polynomials of arbitrary positive degree. Moreover, every complex infinite-dimensional separable Banach space with an unconditional Schauder decomposition and every complex Frèchet space with an unconditional basis support chaotic and frequently hypercyclic polynomials of arbitrary positive degree. We also study distributional chaos for polynomials and show that every infinite-dimensional separable Banach space supports polynomials of arbitrary positive degree that have a dense distributionally scrambled linear manifold. © 2013 Nilson C. Bernardes Jr. and Alfredo Peris.The present work was done while the first author was visiting the Departament de Matematica Aplicada at Universitat Politecnica de Valencia (Spain). The first author is very grateful for the hospitality. The first author was supported in part by CAPES: Bolsista, Project no. BEX 4012/11-9. The second author was supported in part by MEC and FEDER, Project MTM2010-14909, and by GVA, Projects PROMETEO/2008/101 and PROMETEOII/2013/013.Bernardes, NC.; Peris Manguillot, A. (2013). On the existence of polynomials with chaotic behaviour. Journal of Function Spaces and Applications. 2013(320961). https://doi.org/10.1155/2013/320961S2013320961Bayart, F., & Matheron, E. (2009). Dynamics of Linear Operators. doi:10.1017/cbo9780511581113Grosse-Erdmann, K.-G., & Peris Manguillot, A. (2011). Linear Chaos. Universitext. doi:10.1007/978-1-4471-2170-1Rolewicz, S. (1969). On orbits of elements. Studia Mathematica, 32(1), 17-22. doi:10.4064/sm-32-1-17-22Herzog, G. (1992). On linear operators having supercyclic vectors. Studia Mathematica, 103(3), 295-298. doi:10.4064/sm-103-3-295-298Ansari, S. I. (1997). Existence of Hypercyclic Operators on Topological Vector Spaces. Journal of Functional Analysis, 148(2), 384-390. doi:10.1006/jfan.1996.3093Bernal-González, L. (1999). Proceedings of the American Mathematical Society, 127(04), 1003-1011. doi:10.1090/s0002-9939-99-04657-2Bonet, J., & Peris, A. (1998). Hypercyclic Operators on Non-normable Fréchet Spaces. Journal of Functional Analysis, 159(2), 587-595. doi:10.1006/jfan.1998.3315Bonet, 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/33.2.196Shkarin, S. (2008). On the spectrum of frequently hypercyclic operators. Proceedings of the American Mathematical Society, 137(01), 123-134. doi:10.1090/s0002-9939-08-09655-xDe la Rosa, M., Frerick, L., Grivaux, S., & Peris, A. (2011). Frequent hypercyclicity, chaos, and unconditional Schauder decompositions. Israel Journal of Mathematics, 190(1), 389-399. doi:10.1007/s11856-011-0210-6Bernardes, N. C. (1998). ON ORBITS OF POLYNOMIAL MAPS IN BANACH SPACES. Quaestiones Mathematicae, 21(3-4), 311-318. doi:10.1080/16073606.1998.9632049Bernardes Jr., N. C. (1998). Proceedings of the American Mathematical Society, 126(10), 3037-3045. doi:10.1090/s0002-9939-98-04483-9Dineen, S. (1999). Complex Analysis on Infinite Dimensional Spaces. Springer Monographs in Mathematics. doi:10.1007/978-1-4471-0869-6Peris, A. (2001). Proceedings of the American Mathematical Society, 129(12), 3759-3761. doi:10.1090/s0002-9939-01-06274-8ARON, R. M., & MIRALLES, A. (2008). CHAOTIC POLYNOMIALS IN SPACES OF CONTINUOUS AND DIFFERENTIABLE FUNCTIONS. Glasgow Mathematical Journal, 50(2), 319-323. doi:10.1017/s0017089508004229Peris, A. (2003). Chaotic polynomials on Banach spaces. Journal of Mathematical Analysis and Applications, 287(2), 487-493. doi:10.1016/s0022-247x(03)00547-xMARTÍNEZ-GIMÉNEZ, F., & PERIS, A. (2010). CHAOTIC POLYNOMIALS ON SEQUENCE AND FUNCTION SPACES. International Journal of Bifurcation and Chaos, 20(09), 2861-2867. doi:10.1142/s0218127410027416Martínez-Giménez, F., & Peris, A. (2009). Existence of hypercyclic polynomials on complex Fréchet spaces. Topology and its Applications, 156(18), 3007-3010. doi:10.1016/j.topol.2009.02.010Bès, J., & Peris, A. (2007). Disjointness in hypercyclicity. Journal of Mathematical Analysis and Applications, 336(1), 297-315. doi:10.1016/j.jmaa.2007.02.043Bernardes, N. C., Bonilla, A., Müller, V., & Peris, A. (2013). Distributional chaos for linear operators. Journal of Functional Analysis, 265(9), 2143-2163. doi:10.1016/j.jfa.2013.06.019Le�n-Saavedra, F., & M�ller, V. (2004). Rotations of Hypercyclic and Supercyclic Operators. Integral Equations and Operator Theory, 50(3), 385-391. doi:10.1007/s00020-003-1299-8Grosse-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.2010.25Li, T.-Y., & Yorke, J. A. (1975). Period Three Implies Chaos. The American Mathematical Monthly, 82(10), 985. doi:10.2307/2318254Schweizer, B., & Smital, J. (1994). Measures of Chaos and a Spectral Decomposition of Dynamical Systems on the Interval. Transactions of the American Mathematical Society, 344(2), 737. doi:10.2307/2154504Bermúdez, T., Bonilla, A., Martínez-Giménez, F., & Peris, A. (2011). Li–Yorke and distributionally chaotic operators. Journal of Mathematical Analysis and Applications, 373(1), 83-93. doi:10.1016/j.jmaa.2010.06.011Hou, B., Cui, P., & Cao, Y. (2010). Chaos for Cowen-Douglas operators. Proceedings of the American Mathematical Society, 138(03), 929-929. doi:10.1090/s0002-9939-09-10046-1Martínez-Giménez, F., Oprocha, P., & Peris, A. (2009). Distributional chaos for backward shifts. Journal of Mathematical Analysis and Applications, 351(2), 607-615. doi:10.1016/j.jmaa.2008.10.049Schenke, A., & Shkarin, S. (2013). Hypercyclic operators on countably dimensional spaces. Journal of Mathematical Analysis and Applications, 401(1), 209-217. doi:10.1016/j.jmaa.2012.11.013BONET, J., FRERICK, L., PERIS, A., & WENGENROTH, J. (2005). TRANSITIVE AND HYPERCYCLIC OPERATORS ON LOCALLY CONVEX SPACES. Bulletin of the London Mathematical Society, 37(02), 254-264. doi:10.1112/s0024609304003698Shkarin, S. (2012). Hypercyclic operators on topological vector spaces. Journal of the London Mathematical Society, 86(1), 195-213. doi:10.1112/jlms/jdr08

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

Nonlocal operators are chaotic

[EN] We characterize for the first time the chaotic behavior of nonlocal operators that come from a broad class of time-stepping schemes of approximation for fractional differential operators. For that purpose, we use criteria for chaos of Toeplitz operators in Lebesgue spaces of sequences. Surprisingly, this characterization is proved to be-in some cases-dependent of the fractional order of the operator and the step size of the scheme.C. Lizama is partially supported by FONDECYT (Grant No. 1180041) and DICYT, Universidad de Santiago de Chile, USACH. M. Murillo-Arcila is supported by MICINN and FEDER, Projects MTM2016-75963-P and PID2019-105011GB-I00, and by Generalitat Valenciana, Project GVA/2018/110. A. Peris is supported by MICINN and FEDER, Projects MTM2016-75963-P and PID2019-105011GB-I00, and by Generalitat Valenciana, Project PROMETEO/2017/102.Lizama, C.; Murillo Arcila, M.; Peris Manguillot, A. (2020). Nonlocal operators are chaotic. 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Mean Li-Yorke chaos in Banach spaces

[EN] We investigate the notion of mean Li-Yorke chaos for operators on Banach spaces. We show that it differs from the notion of distributional chaos of type 2, contrary to what happens in the context of topological dynamics on compact metric spaces. We prove that an operator is mean Li-Yorke chaotic if and only if it has an absolutely mean irregular vector. As a consequence, absolutely Cesaro bounded operators are never mean Li-Yorke chaotic. Dense mean Li-Yorke chaos is shown to be equivalent to the existence of a dense (or residual) set of absolutely mean irregular vectors. As a consequence, every mean Li-Yorke chaotic operator is densely mean Li-Yorke chaotic on some infinite-dimensional closed invariant subspace. A (Dense) Mean Li-Yorke Chaos Criterion and a sufficient condition for the existence of a dense absolutely mean irregular manifold are also obtained. Moreover, we construct an example of an invertible hypercyclic operator T such that every nonzero vector is absolutely mean irregular for both T and T-1. Several other examples are also presented. Finally, mean Li-Yorke chaos is also investigated for C-0-semigroups of operators on Banach spaces.This work was partially done on a visit of the first author to the Institut Universitari de Matematica Pura i Aplicada at Universitat Politecnica de Valencia, and he is very grateful for the hospitality and support. The first author was partially supported by project #304207/2018-7 of CNPq and by grant #2017/22588-0 of Sao Paulo Research Foundation (FAPESP). The second and third authors were supported by MINECO, Project MTM2016-75963-P. The third author was also supported by Generalitat Valenciana, Project PROMETEO/2017/102. We thank Frederic Bayart for providing us Theorem 27, which answers a previous question of us. We also thank the referee whose careful comments produced an improvement in the presentation of the article.Bernardes, NCJ.; Bonilla, A.; Peris Manguillot, A. (2020). Mean Li-Yorke chaos in Banach spaces. Journal of Functional Analysis. 278(3):1-31. https://doi.org/10.1016/j.jfa.2019.108343S1312783Albanese, A., Barrachina, X., Mangino, E. M., & Peris, A. (2013). Distributional chaos for strongly continuous semigroups of operators. Communications on Pure and Applied Analysis, 12(5), 2069-2082. doi:10.3934/cpaa.2013.12.2069Barrachina, X., & Conejero, J. A. (2012). Devaney Chaos and Distributional Chaos in the Solution of Certain Partial Differential Equations. Abstract and Applied Analysis, 2012, 1-11. doi:10.1155/2012/457019Barrachina, X., & Peris, A. (2012). Distributionally chaotic translation semigroups. Journal of Difference Equations and Applications, 18(4), 751-761. doi:10.1080/10236198.2011.625945Bayart, F., & Grivaux, S. (2006). Frequently hypercyclic operators. Transactions of the American Mathematical Society, 358(11), 5083-5117. doi:10.1090/s0002-9947-06-04019-0BAYART, F., & RUZSA, I. Z. (2013). Difference sets and frequently hypercyclic weighted shifts. Ergodic Theory and Dynamical Systems, 35(3), 691-709. doi:10.1017/etds.2013.77Bermúdez, T., Bonilla, A., Martínez-Giménez, F., & Peris, A. (2011). Li–Yorke and distributionally chaotic operators. Journal of Mathematical Analysis and Applications, 373(1), 83-93. doi:10.1016/j.jmaa.2010.06.011Bernal-González, L., & Bonilla, A. (2016). Order of growth of distributionally irregular entire functions for the differentiation operator. Complex Variables and Elliptic Equations, 61(8), 1176-1186. doi:10.1080/17476933.2016.1149820Bernardes, N. C., Bonilla, A., Müller, V., & Peris, A. (2013). Distributional chaos for linear operators. Journal of Functional Analysis, 265(9), 2143-2163. doi:10.1016/j.jfa.2013.06.019BERNARDES, N. C., BONILLA, A., MÜLLER, V., & PERIS, A. (2014). Li–Yorke chaos in linear dynamics. Ergodic Theory and Dynamical Systems, 35(6), 1723-1745. doi:10.1017/etds.2014.20Bernardes, N. C., Peris, A., & Rodenas, F. (2017). Set-Valued Chaos in Linear Dynamics. Integral Equations and Operator Theory, 88(4), 451-463. doi:10.1007/s00020-017-2394-6Bernardes, N. C., Bonilla, A., Peris, A., & Wu, X. (2018). Distributional chaos for operators on Banach spaces. Journal of Mathematical Analysis and Applications, 459(2), 797-821. doi:10.1016/j.jmaa.2017.11.005Bès, J., Menet, Q., Peris, A., & Puig, Y. (2015). Recurrence properties of hypercyclic operators. Mathematische Annalen, 366(1-2), 545-572. doi:10.1007/s00208-015-1336-3Conejero, J. A., Müller, V., & Peris, A. (2007). Hypercyclic behaviour of operators in a hypercyclic C0-semigroup. Journal of Functional Analysis, 244(1), 342-348. doi:10.1016/j.jfa.2006.12.008Alberto Conejero, J., Rodenas, F., & Trujillo, M. (2015). Chaos for the Hyperbolic Bioheat Equation. Discrete & Continuous Dynamical Systems - A, 35(2), 653-668. doi:10.3934/dcds.2015.35.653Downarowicz, T. (2013). Positive topological entropy implies chaos DC2. Proceedings of the American Mathematical Society, 142(1), 137-149. doi:10.1090/s0002-9939-2013-11717-xFeldman, N. S. (2002). Hypercyclicity and supercyclicity for invertible bilateral weighted shifts. Proceedings of the American Mathematical Society, 131(2), 479-485. doi:10.1090/s0002-9939-02-06537-1Foryś-Krawiec, M., Oprocha, P., & Štefánková, M. (2017). Distributionally chaotic systems of type 2 and rigidity. Journal of Mathematical Analysis and Applications, 452(1), 659-672. doi:10.1016/j.jmaa.2017.02.056Garcia-Ramos, F., & Jin, L. (2016). Mean proximality and mean Li-Yorke chaos. Proceedings of the American Mathematical Society, 145(7), 2959-2969. doi:10.1090/proc/13440Grivaux, S., & Matheron, É. (2014). Invariant measures for frequently hypercyclic operators. Advances in Mathematics, 265, 371-427. doi:10.1016/j.aim.2014.08.002Hou, B., Cui, P., & Cao, Y. (2009). Chaos for Cowen-Douglas operators. Proceedings of the American Mathematical Society, 138(3), 929-936. doi:10.1090/s0002-9939-09-10046-1Huang, W., Li, J., & Ye, X. (2014). Stable sets and mean Li–Yorke chaos in positive entropy systems. Journal of Functional Analysis, 266(6), 3377-3394. doi:10.1016/j.jfa.2014.01.005León-Saavedra, F. (2002). Operators with hypercyclic Cesaro means. Studia Mathematica, 152(3), 201-215. doi:10.4064/sm152-3-1LI, J., TU, S., & YE, X. (2014). Mean equicontinuity and mean sensitivity. Ergodic Theory and Dynamical Systems, 35(8), 2587-2612. doi:10.1017/etds.2014.41Martínez-Giménez, F., Oprocha, P., & Peris, A. (2009). Distributional chaos for backward shifts. Journal of Mathematical Analysis and Applications, 351(2), 607-615. doi:10.1016/j.jmaa.2008.10.049Martínez-Giménez, F., Oprocha, P., & Peris, A. (2012). Distributional chaos for operators with full scrambled sets. Mathematische Zeitschrift, 274(1-2), 603-612. doi:10.1007/s00209-012-1087-8Menet, Q. (2017). Linear chaos and frequent hypercyclicity. Transactions of the American Mathematical Society, 369(7), 4977-4994. doi:10.1090/tran/6808Müller, V., & Vrs˘ovský, J. (2009). Orbits of Linear Operators Tending to Infinity. Rocky Mountain Journal of Mathematics, 39(1). doi:10.1216/rmj-2009-39-1-219Wu, X. (2013). Li–Yorke chaos of translation semigroups. Journal of Difference Equations and Applications, 20(1), 49-57. doi:10.1080/10236198.2013.809712Wu, X., Oprocha, P., & Chen, G. (2016). On various definitions of shadowing with average error in tracing. Nonlinearity, 29(7), 1942-1972. doi:10.1088/0951-7715/29/7/1942Wu, X., Wang, L., & Chen, G. (2017). Weighted backward shift operators with invariant distributionally scrambled subsets. Annals of Functional Analysis, 8(2), 199-210. doi:10.1215/20088752-3802705Yin, Z., & Yang, Q. (2017). Distributionally n-Scrambled Set for Weighted Shift Operators. Journal of Dynamical and Control Systems, 23(4), 693-708. doi:10.1007/s10883-017-9359-6Yin, Z., & Yang, Q. (2017). Distributionally n-chaotic dynamics for linear operators. Revista Matemática Complutense, 31(1), 111-129. doi:10.1007/s13163-017-0226-