Mean Li-Yorke chaos in Banach spaces

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 Ces\`aro 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.Comment: 26 page

Li-Yorke Chaos for Composition Operators on LpL^p-Spaces

Li-Yorke chaos is a popular and well-studied notion of chaos. Several simple and useful characterizations of this notion of chaos in the setting of linear dynamics were obtained recently. In this note we show that even simpler and more useful characterizations of Li-Yorke chaos can be given in the special setting of composition operators on $L^p$ spaces. As a consequence we obtain a simple characterization of weighted shifts which are Li-Yorke chaotic. We give numerous examples to show that our results are sharp

Probing quantum fluctuation theorems in engineered reservoirs

Fluctuation Theorems are central in stochastic thermodynamics, as they allow for quantifying the irreversibility of single trajectories. Although they have been experimentally checked in the classical regime, a practical demonstration in the framework of quantum open systems is still to come. Here we propose a realistic platform to probe fluctuation theorems in the quantum regime. It is based on an effective two-level system coupled to an engineered reservoir, that enables the detection of the photons emitted and absorbed by the system. When the system is coherently driven, a measurable quantum component in the entropy production is evidenced. We quantify the error due to photon detection inefficiency, and show that the missing information can be efficiently corrected, based solely on the detected events. Our findings provide new insights into how the quantum character of a physical system impacts its thermodynamic evolution.Comment: 9 pages, 4 figure

Plurality Voting: the statistical laws of democracy in Brazil

We explore the statistical laws behind the plurality voting system by investigating the election results for mayor held in Brazil in 2004. Our analysis indicate that the vote partition among mayor candidates of the same city tends to be "polarized" between two candidates, a phenomenon that can be closely described by means of a simple fragmentation model. Complex concepts like "government continuity" and "useful vote" can be identified and even statistically quantified through our approach.Comment: 4 pages, 4 figure

An Unusual Antagonistic Pleiotropy in the Penna Model for Biological Ageing

We combine the Penna Model for biological aging, which is based on the mutation-accumulation theory, with a sort of antagonistic pleiotropy. We show that depending on how the pleiotropy is introduced, it is possible to reproduce both the humans mortality, which increases exponentially with age, and fruitfly mortality, which decelerates at old ages, allowing the appearance of arbitrarily old Methuselah's.Comment: 8 pages, 3 figures, to appear in Physica

Immunization and Aging: a Learning Process in the Immune Network

The immune system can be thought as a complex network of different interacting elements. A cellular automaton, defined in shape-space, was recently shown to exhibit self-regulation and complex behavior and is, therefore, a good candidate to model the immune system. Using this model to simulate a real immune system we find good agreement with recent experiments on mice. The model exhibits the experimentally observed refractory behavior of the immune system under multiple antigen presentations as well as loss of its plasticity caused by aging.Comment: 4 latex pages, 3 postscript figures attached. To be published in Physical Review Letters (Tentatively scheduled for 5th Oct. issue

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-

Dynamics of Helping Behavior and Networks in a Small World

To investigate an effect of social interaction on the bystanders' intervention in emergency situations a rescue model was introduced which includes the effects of the victim's acquaintance with bystanders and those among bystanders from a network perspective. This model reproduces the experimental result that the helping rate (success rate in our model) tends to decrease although the number of bystanders $k$ increases. And the interaction among homogeneous bystanders results in the emergence of hubs in a helping network. For more realistic consideration it is assumed that the agents are located on a one-dimensional lattice (ring), then the randomness $p \in [0,1]$ is introduced: the $kp$ random bystanders are randomly chosen from a whole population and the $k-kp$ near bystanders are chosen in the nearest order to the victim. We find that there appears another peak of the network density in the vicinity of $k=9$ and $p=0.3$ due to the cooperative and competitive interaction between the near and random bystanders.Comment: 13 pages, 8 figure

Obtenção de Imagens de Algas por Microscopia de Força Atômica.

bitstream/CNPDIA/10457/1/CT63_2004.pd

Set-Valued Chaos in Linear Dynamics

[EN] We study several notions of chaos for hyperspace dynamics associated to continuous linear operators. More precisely, we consider a continuous linear operator on a topological vector space X, and the natural hyperspace extensions and of T to the spaces of compact subsets of X and of convex compact subsets of X, respectively, endowed with the Vietoris topology. We show that, when X is a complete locally convex space (respectively, a locally convex space), then Devaney chaos (respectively, topological ergodicity) is equivalent for the maps T, and . Also, under very general conditions, we obtain analogous equivalences for Li-Yorke chaos. Finally, some remarks concerning the topological transitivity and weak mixing properties are included, extending results in Banks (Chaos Solitons Fractals 25(3):681-685, 2005) and Peris (Chaos Solitons Fractals 26(1):19-23, 2005).The first author was partially supported by CNPq (Brazil) and by the EBW+ Project (Erasmus Mundus Programme). The second and third authors were supported by MINECO, Projects MTM2013-47093-P and MTM2016-75963-P. The second author was partially supported by GVA, Project PROMETEOII/2013/013.Bernardes, NCJ.; Peris Manguillot, A.; Ródenas Escribá, FDA. (2017). Set-Valued Chaos in Linear Dynamics. Integral Equations and Operator Theory. 88(4):451-463. https://doi.org/10.1007/s00020-017-2394-6S451463884Banks, J.: Chaos for induced hyperspace maps. Chaos Solitons Fractals 25(3), 681–685 (2005)Bauer, W., Sigmund, K.: Topological dynamics of transformations induced on the space of probability measures. Monatsh. Math. 79, 81–92 (1975)Bayart, F., Matheron, É.: Hypercyclic operators failing the hypercyclicity criterion on classical Banach spaces. J. Funct. Anal. 250(2), 426–441 (2007)Bayart, F., Matheron, É.: Dynamics of Linear Operators. Cambridge University Press, Cambridge (2009)Bermúdez, T., Bonilla, A., Martínez-Giménez, F., Peris, A.: Li-Yorke and distributionally chaotic operators. J. Math. Anal. Appl. 373(1), 83–93 (2011)Bernardes Jr., N.C., Bonilla, A., Müller, V., Peris, A.: Li-Yorke chaos in linear dynamics. Ergod. Theory Dyn. Syst. 35(6), 1723–1745 (2015)Bernardes Jr., N.C., Vermersch, R.M.: Hyperspace dynamics of generic maps of the Cantor space. Can. J. Math. 67(2), 330–349 (2015)Bès, J., Menet, Q., Peris, A., Puig, Y.: Strong transitivity properties for operators. Preprint ( arXiv:1703.03724 )Bès, J., Peris, A.: Hereditarily hypercyclic operators. J. Funct. Anal. 167(1), 94–112 (1999)Bonet, J., Frerick, L., Peris, A., Wengenroth, J.: Transitive and hypercyclic operators on locally convex spaces. Bull. Lond. Math. Soc. 37, 254–264 (2005)de la Rosa, M., Read, C.: A hypercyclic operator whose direct sum $$T \oplus T$$ T ⊕ T is not hypercyclic. J. Oper. Theory 61(2), 369–380 (2009)Fedeli, A.: On chaotic set-valued discrete dynamical systems. Chaos Solitons Fractals 23(4), 1381–1384 (2005)Fu, H., Xing, Z.: Mixing properties of set-valued maps on hyperspaces via Furstenberg families. Chaos Solitons Fractals 45(4), 439–443 (2012)Furstenberg, H.: Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory 1, 1–49 (1967)Grivaux, S.: Hypercyclic operators, mixing operators, and the bounded steps problem. J. Oper. Theory 54, 147–168 (2005)Grosse-Erdmann, K.-G., Peris, A.: Weakly mixing operators on topological vector spaces. Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 104(2), 413–426 (2010)Grosse-Erdmann, K.-G., Peris, A.: Linear Chaos. Universitext, Springer-Verlag, Berlin (2011)Guirao, J.L.G., Kwietniak, D., Lampart, M., Oprocha, P., Peris, A.: Chaos on hyperspaces. Nonlinear Anal. 71(1–2), 1–8 (2009)Hernández, P., King, J., Méndez, H.: Compact sets with dense orbit in $$2^X$$ 2 X . Topol. Proc. 40, 319–330 (2012)Herzog, G., Lemmert, R.: On universal subsets of Banach spaces. Math. Z. 229(4), 615–619 (1998)Hou, B., Tian, G., Shi, L.: Some dynamical properties for linear operators. Ill. J. Math. 53(3), 857–864 (2009)Hou, B., Tian, G., Zhu, S.: Approximation of chaotic operators. J. Oper. Theory 67(2), 469–493 (2012)Illanes, A., Nadler Jr., S.B.: Hyperspaces: Fundamentals and Recent Advances. Marcel Dekker Inc, New York (1999)Kupka, J.: On Devaney chaotic induced fuzzy and set-valued dynamical systems. Fuzzy Sets Syst. 177(1), 34–44 (2011)Kwietniak, D., Oprocha, P.: Topological entropy and chaos for maps induced on hyperspaces. Chaos Solitons Fractals 33(1), 76–86 (2007)Li, J., Yan, K., Ye, X.: Recurrence properties and disjointness on the induced spaces. Discrete Contin. Dyn. Syst. 35(3), 1059–1073 (2015)Liao, G., Wang, L., Zhang, Y.: Transitivity, mixing and chaos for a class of set-valued mappings. Sci. China Ser. A Math. 49(1), 1–8 (2006)Liu, H., Shi, E., Liao, G.: Sensitivity of set-valued discrete systems. Nonlinear Anal. 71(12), 6122–6125 (2009)Liu, H., Lei, F., Wang, L.: Li-Yorke sensitivity of set-valued discrete systems, J. Appl. Math. (2013). Article number: 260856Peris, A.: Set-valued discrete chaos. Chaos Solitons Fractals 26(1), 19–23 (2005)Peris, A., Saldivia, L.: Syndetically hypercyclic operators. Integral Equ. Oper. Theory 51, 275–281 (2005)Román-Flores, H.: A note on transitivity in set-valued discrete systems. Chaos Solitons Fractals 17(1), 99–104 (2003)Rudin, W.: Functional Analysis. International Series in Pure and Applied Mathematics, 2nd edn. McGraw-Hill Inc, New York (1991)Salas, H.: A hypercyclic operator whose adjoint is also hypercyclic. Proc. Am. Math. Soc. 112(3), 765–770 (1991)Shkarin, S.: Hypercyclic operators on topological vector spaces. J. Lond. Math. Soc. 86, 195–213 (2012)Wang, Y., Wei, G.: Characterizing mixing, weak mixing and transitivity of induced hyperspace dynamical systems. Topol. Appl. 155(1), 56–68 (2007)Wang, Y., Wei, G., Campbell, W.H.: Sensitive dependence on initial conditions between dynamical systems and their induced hyperspace dynamical systems. Topol. Appl. 156(4), 803–811 (2009)Wu, Y., Xue, X., Ji, D.: Linear transitivity on compact connected hyperspace dynamics. Dyn. Syst. Appl. 21(4), 523–534 (2012)Wu, X., Wang, J., Chen, G.: F-sensitivity and multi-sensitivity of hyperspatial dynamical systems. J. Math. Anal. Appl. 429(1), 16–26 (2015