Translation-invariant probability measures on entire functions

  • Lev Buhovsky
  • Adi GlücksamEmail author
  • Alexander Logunov
  • Mikhail Sodin


We study non-trivial translation-invariant probability measures on the space of entire functions of one complex variable. The existence (and even an abundance) of such measures was proven by Benjamin Weiss. Answering Weiss’ question, we find a relatively sharp lower bound for the growth of entire functions in the support of such measures. The proof of this result consists of two independent parts: the proof of the lower bound and the construction, which yields its sharpness. Each of these parts combines various tools (both classical and new) from the theory of entire and subharmonic functions and from the ergodic theory. We also prove several companion results, which concern the decay of the tails of non-trivial translation-invariant probability measures on the space of entire functions and the growth of locally uniformly recurrent entire and meromorphic functions.


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  1. [1]
    M. E. Becker, Multiparameter groups of measure-preserving tranformations: a simple proof of Wiener’s ergodic theorem, Ann. Prob. 9 (1981), 504–509.CrossRefGoogle Scholar
  2. [2]
    T. Carleman, Extension d’un théorème de Liouville, Acta Math. 48 (1926), 363–366.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Y. Domar, On the existence of a largest subharmonic minorant of a given function, Ark. Mat. 3 (1957), 429–440.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Y. Domar, Uniform boundedness in families related to subharmonic functions, J. London Math. Soc. (2) 38 (1988), 485–491.MathSciNetCrossRefGoogle Scholar
  5. [5]
    M. Einsiedler and Th. Ward, Ergodic Theory with a View Towards Number Theory, Springer, London, 2011.zbMATHGoogle Scholar
  6. [6]
    A. A. Goldberg and I. V. Ostrovskii, Value Distribution of Meromorphic Functions, American Mathematical Society, Providence, RI, 2008.CrossRefGoogle Scholar
  7. [7]
    L. Hormander, Notions of Convexity, Birkhäuser, Boston, MA, 2007.zbMATHGoogle Scholar
  8. [8]
    B. Tsirelson, Divergence of a stationary random vector field can be always positive (a Weiss’ phenomenon), arXiv:0709.1270[math.PR].Google Scholar
  9. [9]
    B. Weiss, Measurable entire functions, Ann. Numer. Math. 4 (1997), 599–605.MathSciNetzbMATHGoogle Scholar

Copyright information

© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  • Lev Buhovsky
    • 1
  • Adi Glücksam
    • 1
    Email author
  • Alexander Logunov
    • 2
    • 3
  • Mikhail Sodin
    • 1
  1. 1.School of MathematicsTel Aviv UniversityTel AvivIsrael
  2. 2.School of MathematicsTel Aviv UniversityTel AvivIsrael
  3. 3.Chebyshev LaboratorySt. Petersburg State UniversitySaint PetersburgRussia

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