Let $(X,B,P)$ be a standard probability space. Let $T:C\rightarrow PPT(X)$ be a free action of the complex plane on the space $(X,B,P)$. We say that the function $F:X\rightarrow C$ is measurably entire if it is measurable and for $P$-a.e $x$ the function $F_x(z):=F(T_zx)$ is entire.
B. Weiss showed in '97 that for every free $C$ action there exists a non-constant measurably entire function. In the talk I will present upper and lower bounds for the growth of such functions.
The talk is partly based on a joint work with L. Buhovsky, A.Logunov, and M. Sodin.