Invariant probability measures on holomorphic functions on the unit disk
Abstract
We study probability measures on the space of holomorphic functions on the unit disk that are invariant under pre-composition by conformal automorphisms and give no mass to constant functions. We determine the slowest possible growth of functions supported by such measures. Let Mf(R) denote the maximum modulus of f on the hyperbolic ball of radius R centered at the origin. We show that given a non-decreasing function G:[1,∞)→(0,∞), there exists an Aut(𝔻)-invariant probability measure μ on Hol(𝔻, ℂ) ∖ {constant functions } such that log+ Mf(R)=Of(G(R)) for μ almost every f if and only if ∫1∞ (R/G(R)) dR <∞. For the lower bound, we use a mass transport argument to show that almost every trajectory of the gradient flow of log |f| has finite energy. To establish sharpness, we construct holomorphic functions of slow growth on infinite Riemann surfaces.
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BibTeX
@misc{ivrii2026invariant,
title = {{Invariant probability measures on holomorphic functions on the unit disk}},
author = {Oleg Ivrii and Uri Kreitner},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40057},
eprint = {2610.40057},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40057},
}
RIS
TY - UNPB
TI - Invariant probability measures on holomorphic functions on the unit disk
AU - Oleg Ivrii
AU - Uri Kreitner
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40057
UR - https://arxiv.org/abs/2610.40057
AB - We study probability measures on the space of holomorphic functions on the unit disk that are invariant under pre-composition by conformal automorphisms and give no mass to constant functions. We determine the slowest possible growth of functions supported by such measures. Let M_f(R) denote the maximum modulus of f on the hyperbolic ball of radius R centered at the origin. We show that given a non-decreasing function G:[1,∞)→(0,∞), there exists an Aut(𝔻)-invariant probability measure μ on Hol(𝔻, ℂ) ∖ {constant functions } such that log^+ M_f(R)=O_f(G(R)) for μ almost every f if and only if ∫_1^∞ (R/G(R)) dR <∞. For the lower bound, we use a mass transport argument to show that almost every trajectory of the gradient flow of log |f| has finite energy. To establish sharpness, we construct holomorphic functions of slow growth on infinite Riemann surfaces.
ER -