TY - JOUR

T1 - Translation-invariant probability measures on entire functions

AU - Buhovsky, Lev

AU - Glücksam, Adi

AU - Logunov, Alexander

AU - Sodin, Mikhail

N1 - Publisher Copyright:
© 2019, The Hebrew University of Jerusalem.

PY - 2019/10/1

Y1 - 2019/10/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85074782097&partnerID=8YFLogxK

U2 - 10.1007/s11854-019-0067-x

DO - 10.1007/s11854-019-0067-x

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AN - SCOPUS:85074782097

SN - 0021-7670

VL - 139

SP - 307

EP - 339

JO - Journal d'Analyse Mathematique

JF - Journal d'Analyse Mathematique

IS - 1

ER -