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Translation-invariant probability measures on entire functions

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Abstract

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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Correspondence to Adi Glücksam.

Additional information

Supported in part by ISF Grant 1380/13, and by the Alon Fellowship.

Supported in part by ERC Advanced Grant 692616 and ISF Grant 382/15.

Supported in part by ERC Advanced Grant 692616 and ISF Grants 1380/13, 382/15.

Supported in part by ERC Advanced Grant 692616 and ISF Grant 382/15.

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Buhovsky, L., Glücksam, A., Logunov, A. et al. Translation-invariant probability measures on entire functions. JAMA 139, 307–339 (2019). https://doi.org/10.1007/s11854-019-0067-x

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  • DOI: https://doi.org/10.1007/s11854-019-0067-x

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