Hostname: page-component-7c8c6479df-995ml Total loading time: 0 Render date: 2024-03-29T15:27:56.641Z Has data issue: false hasContentIssue false

Hypercyclic composition operators on spaces of real analytic functions

Published online by Cambridge University Press:  07 June 2012

JOSÉ BONET
Affiliation:
Instituto Universitario de Matemática Pura y Aplicada IUMPA, Universitat Politècnica de València, E-46071 Valencia, Spain. e-mail: jbonet@mat.upv.es
PAWEŁ DOMAŃSKI
Affiliation:
Faculty of Mathematics and Comp. Sci., A. Mickiewicz University Poznań, Umultowska 87, 61-614 Poznań, Poland. e-mail: domanski@amu.edu.pl

Abstract

We study the dynamical behaviour of composition operators Cϕ defined on spaces (Ω) of real analytic functions on an open subset Ω of ℝd. We characterize when such operators are topologically transitive, i.e. when for every pair of non-empty open sets there is an orbit intersecting both of them. Moreover, under mild assumptions on the composition operator, we investigate when it is sequentially hypercyclic, i.e., when it has a sequentially dense orbit. If ϕ is a self map on a simply connected complex neighbourhood U of ℝ, U ≠ ℂ, then topological transitivity, hypercyclicity and sequential hypercyclicity of Cϕ:(ℝ) → (ℝ) are equivalent.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Partially supported by MEC and FEDER Project MTM2010-15200.

References

REFERENCES

[1]Abate, M.Iteration Theory of Holomorphic Maps on Taut Manifolds (Mediterranean Press Commenda di Rende 1989).Google Scholar
[2]Bayart, F. and Matheron, E.Dynamics of Linear Operators. Cambridge Tracts in Mathematics, 179 (Cambridge University Press, Cambridge, 2009).CrossRefGoogle Scholar
[3]Bernal–González, L. and Montes–Rodríguez, A.Universal functions for composition operators. Complex Variables 27 (1995), 4756.Google Scholar
[4]Bernal–González, L. and Montes–Rodríguez, A.Non-finite dimensional closed vector spaces of universal functions for composition operators. J. Approx. Theory 82 (1995), 375391.CrossRefGoogle Scholar
[5]Bernal–González, L.Universal entire functions for affine endomorphisms of ℂ. J. Math. Anal. Appl. 305 (2005), 690697.CrossRefGoogle Scholar
[6]Bonet, J. and Domański, P.A note on mean ergodic composition operators on spaces of holomorphic functions. Rev. Real Acad. Sci. Mat. RACSAM 105 (2011), 389396.Google Scholar
[7]Bonet, J. and Domański, P.Power bounded composition operators on spaces of analytic functions, Collect. Math. 62 (2011), 6983.CrossRefGoogle Scholar
[8]Bonet, J. and Peris, A.Hypercyclic operators on non-normable Fréchet spaces. J. Funct. Anal. 159 (1998), 587595.CrossRefGoogle Scholar
[9]Bracci, F. and Poggi–Corradini, P.On Valiron's theorem, in: Proc. Future Trends in Geometric Function Theory. (RNC Workshop Jyväskylä 2003) Rep. Univ. Jyväskylä Dept. Math. Stat. 92 (2003), 3955.Google Scholar
[10]Contreras, M. D. Iteración de funciones analíticas en el disco unidad. Preprint (Universidad de Sevilla, 2009).Google Scholar
[11]Contreras, M. D., Díaz–Madrigal, S. and Pommerenke, Ch.Some remarks on the Abel equation in the unit disk. J. London Math. Soc. (2) 75 (2007), 623634.CrossRefGoogle Scholar
[12]Cowen, C. C. and Ko, E.Hermitian weighted composition operators on H2. Trans. Amer. Math. Soc. 362 (2010), 57715801.CrossRefGoogle Scholar
[13]Cowen, C. C. and MacCluer, B. D.Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics (CRC Press, Boca Raton, 1995).Google Scholar
[14]Cowen, C. C. and MacCluer, B. D.Linear fractional maps of the ball and their composition operators. Acta Math. Sci. (Szeged) 66 (2000), 351376.Google Scholar
[15]Domań>ski, P.Notes on real analytic functions and classical.operators, In: Topics in Complex Analysis and Operator Theory, Blasco, O., Bonet, J., Calabuig, J., Jornet, D. (eds.), Contemporary Math. 561 (2012), 347.CrossRefGoogle Scholar
[16]Domański, P., Goliński, M. and Langenbruch, M.A note on composition operators on spaces of real analytic functions. Ann. Polon. Mat. 103 (2012), 209216.CrossRefGoogle Scholar
[17]Domański, P. and Langenbruch, M.Coherent analytic sets and composition of real analytic functions. J. Reine Angew. Math. 582 (2005), 4159.CrossRefGoogle Scholar
[18]Domański, P. and Vogt, D.The space of real analytic functions has no basis. Studia Math. 142 (2000), 187200.CrossRefGoogle Scholar
[19]Garnett, J. B.Bounded Analytic Functions, revised first edition. (Springer, New York, 2007).Google Scholar
[20]Grosse–Erdmann, K. G.Universal families and hypercyclic operators. Bull. Amer. Math. Soc. 36 (1999), 345381.CrossRefGoogle Scholar
[21]Grosse–Erdmann, K. G. and Mortini, R.Universal functions for composition operators with non-automorphic symbol. J. d'Analyse Math. 107 (2009), 355376.CrossRefGoogle Scholar
[22]Grosse–Erdmann, K. G. and Peris, A.Linear Chaos (Springer, Berlin, 2011).CrossRefGoogle Scholar
[23]Guaraldo, F., Macrí, P. and Tancredi, A., Topics on Real Analytic Spaces (Vieweg, Braunschweig 1986).CrossRefGoogle Scholar
[24]Gunning, R. C.Introduction to Holomorphic Functions of Several Variables (Wadsworth and BrooksCole, Belmont 1990).Google Scholar
[25]Hörmander, L.An Introduction to Complex Analysis in Several Variables, 3rd ed. (North-Holland, Amsterdam 1990).Google Scholar
[26]Kobayashi, S.Hyperbolic Complex Spaces (Springer, Berlin 1998).CrossRefGoogle Scholar
[27]Meise, R. and Vogt, D.Introduction to Functional Analysis (Clarendon Press, Oxford 1997).CrossRefGoogle Scholar
[28]Milnor, J.Dynamics in One Complex Variable (Vieweg, 2006).Google Scholar
[29]Shapiro, J.H.Composition Operators and Classical Function Theory, Universitext: Tracts in Mathematics (Springer-Verlag, New York, 1993).CrossRefGoogle Scholar
[30]Shapiro, J. H. Notes on the dynamics of linear operators, Lecture notes, http://www.mth.msu.edu/~shapiro/Pubvit/Downloads/LinDynamics/LynDynamics.htmlGoogle Scholar
[31]Shkarin, S. Existence theorems in linear chaos. arXiv:0810.1192v2.Google Scholar
[32]Vogt, D.Extension operators for real analytic functions on compact subvarieties of ℝd. J. Reine Angew. Math. 606 (2007), 217233.Google Scholar
[33]Vogt, D.Section spaces of real analytic vector bundles and a theorem of Grothendieck and Poly. In Linear and Non-Linear Theory of Generalized Functions and Its Applications, Banach Center Publ. 88. (Polish Acad. Sci. Inst. Math., Warsaw, 2010), p 315321.CrossRefGoogle Scholar
[34] S. Zajc. Hypercyclicity of composition operators in domains of holomorphy, preprint (2012).Google Scholar