Operators with the specification property

[EN] We study a version of the specification property for linear dynamics. Operators having the specification property are investigated, and relationships with other well known dynamical notions such as mixing, Devaney chaos, and frequent hypercyclicity are obtained.Supported by MEC and FEDER, Project MTM2013-47093-P, and by GVA, Projects PROMETEOII/2013/013 and Project ACOMP/2015/005.Bartoll Arnau, S.; Martínez Jiménez, F.; Peris Manguillot, A. (2016). Operators with the specification property. Journal of Mathematical Analysis and Applications. 436:478-488. https://doi.org/10.1016/j.jmaa.2015.12.004S47848843

Linear dynamical systems

This expository survey is dedicated to recent developments in the area of linear dynamics. Topics include frequent hypercyclicity, U -frequent hypercyclicity, reiterative hypercyclicity, operators of C-type, Li-Yorke and distributional chaos, and hypercyclic algebras

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. Chaos An Interdisciplinary Journal of Nonlinear Science. 30(10):1-8. https://doi.org/10.1063/5.0018408183010Abadias, L., & Miana, P. J. (2018). Generalized Cesàro operators, fractional finite differences and Gamma functions. Journal of Functional Analysis, 274(5), 1424-1465. doi:10.1016/j.jfa.2017.10.010Atici, F. M., & Eloe, P. (2009). Discrete fractional calculus with the nabla operator. Electronic Journal of Qualitative Theory of Differential Equations, (3), 1-12. doi:10.14232/ejqtde.2009.4.3Atıcı, F. M., & Eloe, P. W. (2011). Two-point boundary value problems for finite fractional difference equations. Journal of Difference Equations and Applications, 17(4), 445-456. doi:10.1080/10236190903029241Atici, F. M., & Eloe, P. W. (2008). Initial value problems in discrete fractional calculus. Proceedings of the American Mathematical Society, 137(03), 981-989. doi:10.1090/s0002-9939-08-09626-3Atıcı, F. M., & Şengül, S. (2010). Modeling with fractional difference equations. Journal of Mathematical Analysis and Applications, 369(1), 1-9. doi:10.1016/j.jmaa.2010.02.009Banks, J., Brooks, J., Cairns, G., Davis, G., & Stacey, P. (1992). On Devaney’s Definition of Chaos. The American Mathematical Monthly, 99(4), 332-334. doi:10.1080/00029890.1992.11995856Baranov, A., & Lishanskii, A. (2016). Hypercyclic Toeplitz Operators. Results in Mathematics, 70(3-4), 337-347. doi:10.1007/s00025-016-0527-xBayart, F., & Matheron, E. (2009). Dynamics of Linear Operators. doi:10.1017/cbo9780511581113DELAUBENFELS, R., & EMAMIRAD, H. (2001). Chaos for functions of discrete and continuous weighted shift operators. Ergodic Theory and Dynamical Systems, 21(05). doi:10.1017/s0143385701001675Edelman, M. (2014). Caputo standard α-family of maps: Fractional difference vs. fractional. Chaos: An Interdisciplinary Journal of Nonlinear Science, 24(2), 023137. doi:10.1063/1.4885536Edelman, M. (2015). On the fractional Eulerian numbers and equivalence of maps with long term power-law memory (integral Volterra equations of the second kind) to Grünvald-Letnikov fractional difference (differential) equations. Chaos: An Interdisciplinary Journal of Nonlinear Science, 25(7), 073103. doi:10.1063/1.4922834Edelman, M. (2015). Fractional Maps and Fractional Attractors. Part II: Fractional Difference Caputo α- Families of Maps. The interdisciplinary journal of Discontinuity, Nonlinearity, and Complexity, 4(4), 391-402. doi:10.5890/dnc.2015.11.003Erbe, L., Goodrich, C. S., Jia, B., & Peterson, A. (2016). Survey of the qualitative properties of fractional difference operators: monotonicity, convexity, and asymptotic behavior of solutions. Advances in Difference Equations, 2016(1). doi:10.1186/s13662-016-0760-3Ferreira, R. A. C. (2012). A discrete fractional Gronwall inequality. Proceedings of the American Mathematical Society, 140(5), 1605-1612. doi:10.1090/s0002-9939-2012-11533-3Ferreira, R. A. C. (2013). Existence and uniqueness of solution to some discrete fractional boundary value problems of order less than one. Journal of Difference Equations and Applications, 19(5), 712-718. doi:10.1080/10236198.2012.682577Goodrich, C., & Peterson, A. C. (2015). Discrete Fractional Calculus. doi:10.1007/978-3-319-25562-0Goodrich, C. S. (2012). On discrete sequential fractional boundary value problems. Journal of Mathematical Analysis and Applications, 385(1), 111-124. doi:10.1016/j.jmaa.2011.06.022Goodrich, C. S. (2014). A convexity result for fractional differences. Applied Mathematics Letters, 35, 58-62. doi:10.1016/j.aml.2014.04.013Goodrich, C., & Lizama, C. (2020). A transference principle for nonlocal operators using a convolutional approach: fractional monotonicity and convexity. Israel Journal of Mathematics, 236(2), 533-589. doi:10.1007/s11856-020-1991-2Gray, H. L., & Zhang, N. F. (1988). On a new definition of the fractional difference. Mathematics of Computation, 50(182), 513-529. doi:10.1090/s0025-5718-1988-0929549-2Li, K., Peng, J., & Jia, J. (2012). Cauchy problems for fractional differential equations with Riemann–Liouville fractional derivatives. Journal of Functional Analysis, 263(2), 476-510. doi:10.1016/j.jfa.2012.04.011Lizama, C. (2017). The Poisson distribution, abstract fractional difference equations, and stability. Proceedings of the American Mathematical Society, 145(9), 3809-3827. doi:10.1090/proc/12895Lizama, C. (2015). lp-maximal regularity for fractional difference equations on UMD spaces. Mathematische Nachrichten, 288(17-18), 2079-2092. doi:10.1002/mana.201400326Lizama, C., & Murillo-Arcila, M. (2020). Discrete maximal regularity for volterra equations and nonlocal time-stepping schemes. Discrete & Continuous Dynamical Systems - A, 40(1), 509-528. doi:10.3934/dcds.2020020Martínez-Giménez, F. (2007). Chaos for power series of backward shift operators. Proceedings of the American Mathematical Society, 135(6), 1741-1752. doi:10.1090/s0002-9939-07-08658-3Radwan, A. G., AbdElHaleem, S. H., & Abd-El-Hafiz, S. K. (2016). Symmetric encryption algorithms using chaotic and non-chaotic generators: A review. Journal of Advanced Research, 7(2), 193-208. doi:10.1016/j.jare.2015.07.002Radwan, A. G., Moaddy, K., Salama, K. N., Momani, S., & Hashim, I. (2014). Control and switching synchronization of fractional order chaotic systems using active control technique. Journal of Advanced Research, 5(1), 125-132. doi:10.1016/j.jare.2013.01.003Wu, G.-C., & Baleanu, D. (2013). Discrete fractional logistic map and its chaos. Nonlinear Dynamics, 75(1-2), 283-287. doi:10.1007/s11071-013-1065-7Wu, G.-C., Baleanu, D., & Zeng, S.-D. (2014). Discrete chaos in fractional sine and standard maps. Physics Letters A, 378(5-6), 484-487. doi:10.1016/j.physleta.2013.12.01

Cyclicity of coefficient multipliers: Linear structure

In this paper we characterize various kinds of cyclicity of sequences of coefficient multipliers, which are operators defined on spaces of holomorphic functions. In the case of a single coefficient multiplier we characterize its cyclicity, which contrasts with the fact that such operators are never supercyclic. Moreover, it is proved that for each cyclic function there is a dense part of the linear span of its orbit all of whose vectors are cyclic.Ministerio de Ciencia y TecnologíaPlan Andaluz de Investigación (Junta de Andalucía