TY - JOUR
T1 - Measurable Entire Functions II
AU - Glücksam, Adi
AU - Weiss, Benjamin
N1 - Publisher Copyright:
© The Author(s) 2025. Published by Oxford University Press.
PY - 2025/11/1
Y1 - 2025/11/1
N2 - Let ɛ denote the space of entire functions with the topology of uniform convergence on compact sets. The action of C by translations on ɛ is defined by Tzf(w) = f(w + z). Let U denote the set of entire functions whose orbit under T is dense. Birkhoff showed, in 1929, that U is not empty. One of the problems in the collection by T.-C. Dinh and N. Sibony in 2020 asks whether there exists an invariant probability measure on ɛ whose support is contained in U. We will show how an old construction of the second author can be modified to provide a positive answer to their question. Furthermore, we modify the construction to produce a wealth of ergodic measures on the space of entire functions of several complex variables.
AB - Let ɛ denote the space of entire functions with the topology of uniform convergence on compact sets. The action of C by translations on ɛ is defined by Tzf(w) = f(w + z). Let U denote the set of entire functions whose orbit under T is dense. Birkhoff showed, in 1929, that U is not empty. One of the problems in the collection by T.-C. Dinh and N. Sibony in 2020 asks whether there exists an invariant probability measure on ɛ whose support is contained in U. We will show how an old construction of the second author can be modified to provide a positive answer to their question. Furthermore, we modify the construction to produce a wealth of ergodic measures on the space of entire functions of several complex variables.
UR - https://www.scopus.com/pages/publications/105021526830
U2 - 10.1093/imrn/rnaf330
DO - 10.1093/imrn/rnaf330
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AN - SCOPUS:105021526830
SN - 1073-7928
VL - 2025
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
IS - 21
M1 - rnaf330
ER -