On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple

In this paper we extend the notion of a locally hypercyclic operator to that of a locally hypercyclic tuple of operators. We then show that the class of hypercyclic tuples of operators forms a proper subclass to that of locally hypercyclic tuples of operators. What is rather remarkable is that in every finite dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$, a pair of commuting matrices exists which forms a locally hypercyclic, non-hypercyclic tuple. This comes in direct contrast to the case of hypercyclic tuples where the minimal number of matrices required for hypercyclicity is related to the dimension of the vector space. In this direction we prove that the minimal number of diagonal matrices required to form a hypercyclic tuple on $\mathbb{R}^n$ is $n+1$, thus complementing a recent result due to Feldman.Comment: 15 pages, title changed, section for infinite dimensional spaces adde

Experimental investigation of a simple distortion index utilizing steady-state and dynamic distortions in a Mach 2.5 mixed-compression inlet and turbofan engine

A wind tunnel investigation was conducted to determine the amplitude and spatial distribution of steady-state and dynamic distortion produced in an inlet with 45 percent of the overall supersonic area contraction occurring internally. It was found that the inlet support strut location and/or the overboard bypass flow rate has a significant effect on the spatial distribution of distortion. Because of this effect the majority of the stall points exhibited four-per-revolution patterns of distortion. Data from this test were used to formulate a simple index that combines steady-state and dynamic distortions. Distortion results obtained with this index correlated well with exhaust nozzle area. It is shown that the exhaust nozzle area of a TF30-P-3, as modified for use in this test, can be controlled in a scheme to avoid engine stall. A considerable increase in engine distortion tolerance can be achieved by opening the 7th-stage bleed. The engine exhibited higher tolerance to distortion for multiple patterns of distortion per-revolution than for a one-per-revolution pattern of distortion

J-Class Abelian Semigroups of Matrices on C^n and Hypercyclicity

We give a characterization of hypercyclic finitely generated abelian semigroups of matrices on C^n using the extended limit sets (the J-sets). Moreover we construct for any n\geq 2 an abelian semigroup G of GL(n;C) generated by n + 1 diagonal matrices which is locally hypercyclic but not hypercyclic and such that JG(e_k) = C^n for every k = 1; : : : ; n, where (e_1; : : : ; e_n) is the canonical basis of C^n. This gives a negative answer to a question raised by Costakis and Manoussos.Comment: 10 page

Identical approximative sequence for various notions of universality

AbstractIn this paper, we examine various notions of universality, which have already been proved generic. Our main purpose is to prove that generically they occur simultaneously with the same approximative sequence

Distributional chaos for operators with full scrambled sets

In this article we answer in the negative the question of whether hypercyclicity is sufficient for distributional chaos for a continuous linear operator (we even prove that the mixing property does not suffice). Moreover, we show that an extremal situation is possible: There are (hypercyclic and non-hypercyclic) operators such that the whole space consists, except zero, of distributionally irregular vectors.The research of first and third author was supported by MEC and FEDER, project MTM2010-14909 and by GV, Project PROMETEO/2008/101. The research of second author was supported by the Marie Curie European Reintegration Grant of the European Commission under grant agreement no. PERG08-GA-2010-272297. The financial support of these institutions is hereby gratefully acknowledged. We also want to thank X. Barrachina for pointing out to us a gap in the proof of a previous version of Theorem 3.1.Martínez Jiménez, F.; Oprocha, P.; Peris Manguillot, A. (2013). Distributional chaos for operators with full scrambled sets. Mathematische Zeitschrift. 274(1-2):603-612. https://doi.org/10.1007/s00209-012-1087-8S6036122741-2Banks, J., Brooks, J., Cairns, G., Davis, G., Stacey, P.: On Devaney’s definition of chaos. Am. Math. Monthly 99(4), 332–334 (1992)Barrachina, X., Peris, A.: Distributionally chaotic translation semigroups. J. Differ. Equ. Appl. 18, 751–761 (2012)Beauzamy, B.: Introduction to Operator Theory and Invariant Subspaces. North-Holland, Amsterdam (1988)Bermúdez, T., Bonilla, A., Martínez-Giménez, F., Peris, A.: Li–Yorke and distributionally chaotic operators. J. Math. Anal. Appl. 373, 83–93 (2011)Bayart, F., Matheron, E.: Dynamics of linear operators, vol. 179. Cambridge University Press, London(2009).Costakis, G., Sambarino, M.: Topologically mixing hypercyclic operators. Proc. Am. Math. Soc. 132, 385–389 (2004)Devaney, R.L.: An introduction to chaotic dynamical systems, 2nd edn. Addison-Wesley Studies in Nonlinearity. Addison-Wesley Publishing Company Advanced Book Program. Redwood City (1989)Feldman, N.: Hypercyclicity and supercyclicity for invertible bilateral weighted shifts. Proc. Am. Math. Soc. 131, 479–485 (2003)Grosse-Erdmann, K.-G.: Hypercyclic and chaotic weighted shifts. Studia Math. 139(1), 47–68 (2000)Grosse-Erdmann, K.-G., Peris Manguillot, A.: Linear Chaos. Universitext, Springer, London (2011)Hou, B., Cui, P., Cao, Y.: Chaos for Cowen-Douglas operators. Proc. Am. Math. Soc 138, 929–936 (2010)Hou, B., Tian, G., Shi, L.: Some dynamical properties for linear operators. Ill. J. Math. 53, 857–864 (2009)Li, T.Y., Yorke, J.A.: Period three implies chaos. Am. Math. Monthly 82(10), 985–992 (1975)Martínez-Giménez, F., Oprocha, P., Peris, A.: Distributional chaos for backward shifts. J. Math. Anal. Appl. 351, 607–615 (2009)Müller, V., Peris, A.: A Problem of Beauzamy on Irregular Operators (2011). (Preprint)Oprocha, P.: Distributional chaos revisited. Trans. Am. Math. Soc. 361, 4901–4925 (2009)Oprocha, P.: A quantum harmonic oscillator and strong chaos. J. Phys. A 39(47), 14559–14565 (2006)Schweizer, B., Smítal, J.: Measures of chaos and a spectral decomposition of dynamical systems on the interval. Trans. Am. Math. Soc. 344(2), 737–754 (1994)Wu, X., Zhu, P.: The principal measure of a quantum harmonic oscillator. J. Phys. A 44(505101), 6 (2011

Recurrence properties of hypercyclic operators

[EN] We generalize the notions of hypercyclic operators, U-frequently hypercyclic operators and frequently hypercyclic operators by introducing a new concept in linear dynamics, namely A-hypercyclicity. We then state an A-hypercyclicity criterion, inspired by the hypercyclicity criterion and the frequent hypercyclicity criterion, and we show that this criterion characterizes the A-hypercyclicity for weighted shifts. We also investigate which density properties can the sets N(x, U) = {n is an element of N; T-n x is an element of U} have for a given hypercyclic operator, and we study the new notion of reiteratively hypercyclic operators.This work is supported in part by MEC and FEDER, Project MTM2013-47093-P, and by GVA, Projects PROMETEOII/2013/013 and ACOMP/2015/005. The second author was a postdoctoral researcher of the Belgian FNRS.Bès, JP.; Menet, Q.; Peris Manguillot, A.; Puig-De Dios, Y. (2016). Recurrence properties of hypercyclic operators. Mathematische Annalen. 366(1):545-572. https://doi.org/10.1007/s00208-015-1336-3S5455723661Badea, C., Grivaux, S.: Unimodular eigenvalues, uniformly distributed sequences and linear dynamics. Adv. Math. 211, 766–793 (2007)Bayart, F., Grivaux, S.: Frequently hypercyclic operators. Trans. Amer. Math. Soc. 358, 5083–5117 (2006)Bayart, F., Grivaux, S.: Invariant Gaussian measures for operators on Banach spaces and linear dynamics. Proc. Lond. Math. Soc. 94, 181–210 (2007)Bayart, F., Matheron, É.: Dynamics of linear operators, Cambridge Tracts in Mathematics, 179. Cambridge University Press, Cambridge (2009)Bayart, F., Matheron, É.: (Non-)weakly mixing operators and hypercyclicity sets. Ann. Inst. Fourier 59, 1–35 (2009)Bayart, F., Ruzsa, I.: Difference sets and frequently hypercyclic weighted shifts. Ergodic Theory Dynam. Syst. 35, 691–709 (2015)Bergelson, V.: Ergodic Ramsey Theory- an update, Ergodic Theory of $$\mathbb{Z}^d$$ Z d -actions. Lond. Math. Soc. Lecture Note Ser. 28, 1–61 (1996)Bernal-González, L., Grosse-Erdmann, K.-G.: The Hypercyclicity Criterion for sequences of operators. Studia Math. 157, 17–32 (2003)Bès, J., Peris, A.: Hereditarily hypercyclic operators. J. Funct. Anal. 167, 94–112 (1999)Bonilla, A., Grosse-Erdmann, K.-G.: Frequently hypercyclic operators and vectors. Ergodic Theory Dynam. Syst. 27, 383–404 (2007)Bonilla, A., Grosse-Erdmann, K.-G.: Erratum: Ergodic Theory Dynam. Systems 29, 1993–1994 (2009)Chan, K., Seceleanu, I.: Hypercyclicity of shifts as a zero-one law of orbital limit points. J. Oper. Theory 67, 257–277 (2012)Costakis, G., Sambarino, M.: Topologically mixing hypercyclic operators. Proc. Amer. Math. Soc. 132, 385–389 (2004)Furstenberg, H.: Recurrence in ergodic theory and combinatorial number theory. Princeton University Press, Princeton (1981)Giuliano, R., Grekos, G., Mišík, L.: Open problems on densities II, Diophantine Analysis and Related Fields 2010. AIP Conf. Proc. 1264, 114–128 (2010)Grosse-Erdmann, K.-G.: Hypercyclic and chaotic weighted shifts. Studia Math. 139, 47–68 (2000)Grosse-Erdmann, K.-G., Peris, A.: Frequently dense orbits. C. R. Math. Acad. Sci. Paris 341, 123–128 (2005)Grosse-Erdmann, K.G., Peris, A.: Weakly mixing operators on topological vector spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 104, 413–426 (2010)Grosse-Erdmann, K.G., Peris Manguillot, A.: Linear chaos, Universitext. Springer, London (2011)Menet, Q.: Linear chaos and frequent hypercyclicity. Trans. Amer. Math. Soc. arXiv:1410.7173Puig, Y.: Linear dynamics and recurrence properties defined via essential idempotents of $$\beta {\mathbb{N}}$$ β N (2014) arXiv:1411.7729 (preprint)Salas, H.N.: Hypercyclic weighted shifts. Trans. Amer. Math. Soc. 347, 993–1004 (1995)Salat, T., Toma, V.: A classical Olivier’s theorem and statistical convergence. Ann. Math. Blaise Pascal 10, 305–313 (2003)Shkarin, S.: On the spectrum of frequently hypercyclic operators. Proc. Am. Math. Soc. 137, 123–134 (2009

On a conjecture of D. Herrero concerning hypercyclic operators

We give an affirmative answer to a conjecture of D. Herrero [6]. Let X denote a real or complex locally convex linear space and T : X –> X a continuous linear operator. Suppose that we are given x(1),..., x(n) is an element of X and boolean ORk=1n Orb(T, x(k)) is dense in X. Then Orb(T, x(j)) is dense in X for some j is an element of 1,...,n. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS

Some remarks on universal functions and Taylor series

We derive properties of universal functions and Taylor series in domains of the complex plane. For some of our results we use Baire’s theorem. We also give a constructive proof, avoiding Baire’s theorem, of the existence of universal Taylor series in any arbitrary simply connected domain