Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173



Limit functions of discrete dynamical systems

Authors: H.-P. Beise, T. Meyrath and J. Müller
Journal: Conform. Geom. Dyn. 18 (2014), 56-64
MSC (2010): Primary 37A25, 37F10, 30K99
Published electronically: April 1, 2014
MathSciNet review: 3187620
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In the theory of dynamical systems, the notion of $ \omega $-limit sets of points is classical. In this paper, the existence of limit functions on subsets of the underlying space is treated. It is shown that in the case of topologically mixing systems on appropriate metric spaces $ (X,d)$, the existence of at least one limit function on a compact subset $ A$ of $ X$ implies the existence of plenty of them on many supersets of $ A$. On the other hand, such sets necessarily have to be small in various respects. The results for general discrete systems are applied in the case of Julia sets of rational functions and in particular in the case of the existence of Siegel disks.

References [Enhancements On Off] (What's this?)

  • [1] Ethan Akin, Lectures on Cantor and Mycielski sets for dynamical systems, Chapel Hill Ergodic Theory Workshops, Contemp. Math., vol. 356, Amer. Math. Soc., Providence, RI, 2004, pp. 21-79. MR 2087588 (2005e:37018),
  • [2] Peter Beise and Jürgen Müller, Limit functions of iterates of entire functions on parts of the Julia set, Proc. Amer. Math. Soc. 141 (2013), no. 11, 3929-3933. MR 3091783,
  • [3] Walter Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 151-188. MR 1216719 (94c:30033),
  • [4] Hans Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103-144 (1965). MR 0194595 (33 #2805)
  • [5] Lennart Carleson and Theodore W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1230383 (94h:30033)
  • [6] Alexandre Freire, Artur Lopes, and Ricardo Mañé, An invariant measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), no. 1, 45-62. MR 736568 (85m:58110b),
  • [7] Karl-G. Grosse-Erdmann and Alfredo Peris Manguillot, Linear chaos, Universitext, Springer, London, 2011. MR 2919812
  • [8] Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597 (96e:03057)
  • [9] Steven P. Lalley, Brownian motion and the equilibrium measure on the Julia set of a rational mapping, Ann. Probab. 20 (1992), no. 4, 1932-1967. MR 1188049 (94f:58079)
  • [10] Artur Oscar Lopes, Equilibrium measures for rational maps, Ergodic Theory Dynam. Systems 6 (1986), no. 3, 393-399. MR 863202 (88e:58055),
  • [11] M. Ju. Ljubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems 3 (1983), no. 3, 351-385. MR 741393 (85k:58049),
  • [12] M. Yu. Lyubich, Generic behavior of trajectories of the exponential function, Uspekhi Mat. Nauk 41 (1986), no. 2(248), 199-200 (Russian). MR 842176 (87g:58062)
  • [13] M. Yu. Lyubich, The measurable dynamics of the exponential, Sibirsk. Mat. Zh. 28 (1987), no. 5, 111-127 (Russian). MR 924986 (89d:58071)
  • [14] John Milnor, Dynamics in one complex variable, Friedr. Vieweg & Sohn, Braunschweig, 1999. Introductory lectures. MR 1721240 (2002i:37057)
  • [15] Yûsuke Okuyama and Małgorzata Stawiska, Potential theory and a characterization of polynomials in complex dynamics, Conform. Geom. Dyn. 15 (2011), 152-159. MR 2846305,
  • [16] Thomas Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts, vol. 28, Cambridge University Press, Cambridge, 1995. MR 1334766 (96e:31001)
  • [17] Mary Rees, The exponential map is not recurrent, Math. Z. 191 (1986), no. 4, 593-598. MR 832817 (87g:58063),
  • [18] Walter Rudin, Functional analysis, McGraw-Hill Book Co., New York, 1973. McGraw-Hill Series in Higher Mathematics. MR 0365062 (51 #1315)
  • [19] Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157 (88k:00002)
  • [20] Dierk Schleicher, Dynamics of entire functions, Holomorphic dynamical systems, Lecture Notes in Math., vol. 1998, Springer, Berlin, 2010, pp. 295-339. MR 2648691 (2011h:37070),
  • [21] Norbert Steinmetz, Rational iteration, de Gruyter Studies in Mathematics, vol. 16, Walter de Gruyter & Co., Berlin, 1993. Complex analytic dynamical systems. MR 1224235 (94h:30035)
  • [22] Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York, 1982. MR 648108 (84e:28017)

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2010): 37A25, 37F10, 30K99

Retrieve articles in all journals with MSC (2010): 37A25, 37F10, 30K99

Additional Information

H.-P. Beise
Affiliation: Department of Mathematics, University of Trier, 54286 Trier, Germany

T. Meyrath
Affiliation: University of Luxembourg, Faculte des Sciences 6, rue Richard Coudenhove-Kalergi, L-1359, Luxembourg

J. Müller
Affiliation: University of Trier, Mathematik, Fachbereich IV, 54286 Trier, Germany

Keywords: Julia set, limit set, Siegel disk, universality
Received by editor(s): May 22, 2013
Received by editor(s) in revised form: November 20, 2013, December 30, 2013, and December 31, 2013
Published electronically: April 1, 2014
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society