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

Ergodic and dynamical properties of m-isometries

[EN] An example of a weakly ergodic 3-isometry is provided in [3], we give new examples of weakly ergodic 3-isometries and study numerically hypercyclic m-isometries on finite and infinite dimensional Hilbert spaces. In particular, all weakly ergodic strict 3-isometries on a Hilbert space are weakly numerically hypercyclic. Adjoints of unilateral forward weighted shifts which are strict m-isometries on l(2)(N) are shown to be hypercyclic. (C) 2018 Elsevier Inc. All rights reserved.The first, second and fourth authors were supported by MINECO and FEDER, Project MTM2016-75963-P. The third author was supported by grant No. 17-27844S of GA CR and RVO: 67985840. The fourth author was also supported by Generalitat Valenciana, Project PROMETEO/2017/102.Bermudez, T.; Bonilla, A.; Müller, V.; Peris Manguillot, A. (2019). Ergodic and dynamical properties of m-isometries. Linear Algebra and its Applications. 561:98-112. https://doi.org/10.1016/j.laa.2018.09.0229811256

Cesaro bounded operators in Banach spaces

[EN] We study several notions of boundedness for operators. It is known that any power bounded operator is absolutely Cesaro bounded and strongly Kreiss bounded (in particular, uniformly Kreiss bounded). The converses do not hold in general. In this note, we give examples of topologically mixing (hence, not power bounded) absolutely Cesaro bounded operators on l(p)(N), 1 <= p < infinity, and provide examples of uniformly Kreiss bounded operators which are not absolutely Cesaro bounded. These results complement a few known examples (see [27] and [2]). We also obtain a characterization of power bounded operators which generalizes a result of Van Casteren [32]. In [2] Aleman and Suciu asked if every uniformly Kreiss bounded operator T on a Banach space satisfies that. We solve this question for Hilbert space operators and, moreover, we prove that, if T is absolutely Cesaro bounded on a Banach (Hilbert) space, then parallel to T-n parallel to = o(n) ((parallel to Tn parallel to=o(n12), respectively). As a consequence, every absolutely Cesaro bounded operator on a reflexive Banach space is mean ergodic.The first, second and fourth authors were supported by MINECO and FEDER, Project MTM201675963-P. The third author was supported by grant No. 17-27844S of GA CR and RVO: 67985840. The fourth author was also supported by Generalitat Valenciana, Project PROMETEO/2017/102.Bermúdez, T.; Bonilla, A.; Muller, V.; Peris Manguillot, A. (2020). Cesaro bounded operators in Banach spaces. Journal d Analyse Mathématique. 140(1):187-206. https://doi.org/10.1007/s11854-020-0085-8S1872061401A. Albanese, J. Bonet and W. J. Ricker, Mean ergodic operators in Fréchet spaces, Ann. Acad. Sci. Fenn. Math. 34 (2009), 401–436.A. Aleman and L. Suciu, On ergodic operator means in Banach spaces, Integral Equations Operator Theory 85 (2016), 259–287.I. Assani, Sur les opérateurs à puissances bornées et le théorème ergodique ponctuel dans Lp[0, 1], 1 < p < ∞, Canad. J. Math. 38 (1986), 937–946.M. J. Beltrán-Meneu, Operators on Weighted Spaces of Holomorphic Functions, PhD Thesis, Universitat Politècnica de Valencia, Valencia, Spain, 2014.M. J. Beltrán, J. Bonet and C. Fernández, Classical operators on weighted Banach spaces of entire functions, Proc. Amer. Math. Soc. 141 (2013), 4293–4303.M. J. Beltrán, M.C. Gómez-Collado, E. Jordá and D. Jornet, Mean ergodic composition operators on Banach spaces of holomorphic functions, J. Funct. Anal. 270 (2016), 4369–4385.N. C. Bernardes, Jr., A. Bonilla, V. Müller and A. Peris, Distributional chaos for linear operators, J. Funct. Anal. 265 (2013), 2143–2163.N. C. Bernardes, Jr., A. Bonilla, A. Peris and X. Wu, Distributional chaos for operators in Banach spaces, J. Math. Anal. Appl. 459 (2018), 797–821.J. Bonet, Dynamics of the differentiation operator on weighted spaces of entire functions, Math. Z. 261 (2009), 649–657.Y. Derriennic, On the mean ergodic theorem for Cesaro bounded operators, Colloq. Math. 84/85 (2000), 443–455.Y. Derriennic and M. Lin, On invariant measures and ergodic theorems for positive operators, J. Funct. Anal. 13 (1973), 252–267.R. Émilion, Mean-bounded operators and mean ergodic theorems, J. Funct. Anal. 61 (1985), 1–14.A. Gomilko and J. Zemánek, On the uniform Kreiss resolvent condition, (Russian) Funktsional. Anal. i Prilozhen. 42 (2008), 81–84A. Gomilko and J. Zemánek, English translation in Funct. Anal. Appl. 42 (2008), 230–233.K.-G. Grosse-Erdmann and A. Peris, Linear Chaos, Springer, London, 2011.B. Z. Guo and H. Zwart, On the relation between stability of continuous- and discrete-time evolution equations via the Cayley transform, Integral Equations Operator Theory 54 (2006), 349–383.B. Hou and L. Luo, Some remarks on distributional chaos for bounded linear operators, Turk. J. Math. 39 (2015), 251–258.E. Hille, Remarks on ergodic theorems, Trans. Amer. Math. Soc. 57 (1945), 246–269.I. Kornfeld and W. Kosek, Positive L1operators associated with nonsingular mappings and an example of E. Hille, Colloq. Math. 98 (2003), 63–77.W. Kosek, Example of a mean ergodic L1operator with the linear rate of growth, Colloq. Math. 124 (2011), 15–22.U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin, 1985.C. Lubich and O. Nevanlinna, On resolvent conditions and stability estimates, BIT, 31 (1991), 293–313.C. A. McCarthy, A Strong Resolvent Condition does not Imply Power-Boundedness, Chalmers Institute of Technology and the University of Göteborg, Preprint No. 15 (1971).A. Montes-Rodríguez, J. Sánchez-Álvarez and J. Zemánek, Uniform Abel—Kreiss boundedness and the extremal behavior of the Volterra operator, Proc. London Math. Soc. 91 (2005), 761–788.V. Müller and J. Vrsovsky, Orbits of linear operators tending to infinity, Rocky Mountain J. Math. 39 (2009), 219–230.O. Nevanlinna, Resolvent conditions and powers of operators, Studia Math. 145 (2001), 113–134.J. Neveu, Mathematical Foundations of the Calculus of Probability, Holden-Day, San Francisco, Calif.-London-Amsterdam, 1965.A. L. Shields, On Möbius Bounded operators, Acta Sci. Math. (Szeged) 40 (1978), 371–374.J. C. Strikwerda and B. A. Wade, A survey of the Kreiss matrix theorem for power bounded families of matrices and its extensions, in Linear Operators, Polish Acad. Sci., Warsaw, 1997, pp. 339–360.L. Suciu, Ergodic behaviors of the regular operator means, Banach J. Math. Anal. 11 (2017), 239–265.L. Suciu and J. Zemánek, Growth conditions on Cesàro means of higher order, Acta Sci. Math (Szeged) 79 (2013), 545–581.Y. Tomilov and J. Zemánek, A new way of constructing examples in operator ergodic theory, Math. Proc. Cambridge Philos. Soc. 137 (2004), 209–225.J. A. Van Casteren, Boundedness properties of resolvents and semigroups of operators, in Linear Operators, Polish Acad. Sci. Inst. Math., Warsaw, 1997, pp. 59–74

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-

A dynamic trading rule based on filtered flag pattern recognition for stock market price forecasting

[EN] In this paper we propose and validate a trading rule based on flag pattern recognition, incorporating im- portant innovations with respect to the previous research. Firstly, we propose a dynamic window scheme that allows the stop loss and take profit to be updated on a quarterly basis. In addition, since the flag pat- tern is a trend-following pattern, we have added the EMA indicator to filter trades. This technical analysis indicator is calculated both for 15-min and 1-day timeframes, which enables short and medium terms to be considered simultaneously. We also filter the flags according to the price range on which they are de- veloped and have limited the maximum loss of each trade to 100 points. The proposed methodology was applied to 91,309 intraday observations of the DJIA index, considerably improving the results obtained in the previous proposals and those obtained by the buy & hold strategy, both for profitability and risk, and also after taking into account the transaction costs. These results seem to challenge market efficiency in line with other similar studies, in the specific analysis carried out on the DJIA index and is also limited to the setup considered.The fourth author of this work was partially supported by MINECO, Project MTM2016-75963-P.Arévalo, R.; García, J.; Guijarro, F.; Peris Manguillot, A. (2017). A dynamic trading rule based on filtered flag pattern recognition for stock market price forecasting. Expert Systems with Applications. 81:177-192. https://doi.org/10.1016/j.eswa.2017.03.0281771928

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

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

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