Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-23T20:31:52.462Z Has data issue: false hasContentIssue false
  • English
  • Français

Entropy of transcendental entire functions

Part of: Entire and meromorphic functions, and related topics Complex dynamical systems

Published online by Cambridge University Press:  07 October 2019

ANNA MIRIAM BENINI Open the ORCID record for ANNA MIRIAM BENINI [Opens in a new window] ,
JOHN ERIK FORNÆSS  and
HAN PETERS
ANNA MIRIAM BENINI
Affiliation:
Dipartimento di Scienze Matematiche Fisiche e Informatiche, Università di Parma, Italy email [email protected]
JOHN ERIK FORNÆSS
Affiliation:
Department of Mathematical Sciences, NTNU Trondheim, Norway email [email protected]
HAN PETERS
Affiliation:
Korteweg de Vries Institute for Mathematics, University of Amsterdam, the Netherlands email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that all transcendental entire functions have infinite topological entropy.

Keywords

transcendental dynamicsentropyhyperbolic metric

MSC classification

Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions 30D20: Entire functions, general theory
Secondary: 30D35: Distribution of values, Nevanlinna theory
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 2 , February 2021 , pp. 338 - 348
Check for updates
Copyright
© Cambridge University Press, 2019

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.)

References

Ahlfors, L. V.. Conformal Invariants: Topics in Geometric Function Theory (McGraw-Hill Series in Higher Mathematics) . McGraw-Hill, New York, 1973.Google Scholar
Adler, R. L., Konheim, A. G. and McAndrew, M. H.. Topological entropy. Trans. Amer. Math. Soc. 114 (1965), 309–319.CrossRefGoogle Scholar
Bergweiler, W.. The role of the Ahlfors five islands theorem in complex dynamics. Conform. Geom. Dyn. 4 (2000), 22–34.CrossRefGoogle Scholar
Benini, A. M., Fornæss, J. E. and Peters, H.. Infinite entropy for transcendental entire functions with an omitted value. Acta Math. Vietnam (2018), doi:10.1007/s40306-018-0300-1.Google Scholar
Bowen, R.. Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153 (1971), 401–414.CrossRefGoogle Scholar
Bowen, R.. Erratum to ‘Entropy for group endomorphisms and homogeneous spaces’. Trans. Amer. Math. Soc. 181 (1973), 509–510.Google Scholar
Bowen, R.. Topological entropy for noncompact sets. Trans. Amer. Math. Soc. 184 (1973), 125–136.CrossRefGoogle Scholar
Christensen, J. P. R. and Fischer, P.. Ergodic invariant probability measures and entire functions. Acta Math. Hungar. 73(3) (1996), 213–218.CrossRefGoogle Scholar
Dujardin, R.. Hénon-like mappings in ℂ2 . Amer. J. Math. 126(2) (2004), 439–472.CrossRefGoogle Scholar
Gromov, M.. On the entropy of holomorphic maps. Enseign. Math. (2) 49(3–4) (2003), 217–235.Google Scholar
Hempel, J. A.. The Poincaré metric on the twice punctured plane and the theorems of Landau and Schottky. J. Lond. Math. Soc. (2) 20(3) (1979), 435–445.CrossRefGoogle Scholar
Hasselblatt, B., Nitecki, Z. and Propp, J.. Topological entropy for nonuniformly continuous maps. Discrete Contin. Dyn. Syst. 22(1–2) (2008), 201–213.Google Scholar
Hofer, J. E.. Topological entropy for noncompact spaces. Michigan Math. J. 21 (1974), 235–242.Google Scholar
Jenkins, J. A.. On explicit bounds in Landau’s theorem. II. Canad. J. Math. 33(3) (1981), 559–562.CrossRefGoogle Scholar
Ljubich, M. Ju.. Entropy properties of rational endomorphisms of the Riemann sphere. Ergod. Th. & Dynam. Sys. 3(3) (1983), 351–385.CrossRefGoogle Scholar
Misiurewicz, M. and Przytycki, F.. Topological entropy and degree of smooth mappings. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25(6) (1977), 573–574.Google Scholar
Rippon, P. J. and Stallard, G. M.. Annular itineraries for entire functions. Trans. Amer. Math. Soc. 367(1) (2015), 377–399.CrossRefGoogle Scholar
Wendt, M.. Zufällige Juliamengen und invariante Maße mit maximaler Entropie. PhD Thesis, University of Kiel, 2005, https://macau.uni-kiel.de/receive/dissertation_diss_00001412 (in German).Google Scholar