Random power series generated by ergodic transformations

Abstract:

Generalizing classical studies of power series with sequences of independent random variables as coefficients, we study series of the forms \[ {g_{x,\phi }}(z) = \sum \limits _{n = 0}^\infty {\phi ({T^n}x){z^n}\quad {\text {and}}\quad {f_{x,\phi }}(z) = \sum \limits _{n = 1}^\infty {\phi (x)\phi (Tx) \cdots \phi ({T^{n - 1}}x){z^n},} } \] where $T$ is an ergodic measure-preserving transformation on a probability space $(X,\mathcal {B},\mu )$ and $\phi$ is a measurable complex-valued function which is a.e. nonzero. When ${f_{x,\phi }}$ is entire, its order of growth at infinity measures the speed of divergence of the ergodic averages of $\log |\phi |$. We give examples to show that any order is possible for any $T$ and that different orders are possible for fixed $\phi$. For fixed $T$, the set of $\phi$ which produce infinite order is residual in the subset of ${L^1}(X)$ consisting of those $\phi$ which are a.e. nonzero and produce entire ${f_{x,\phi }}$. As in a theorem of Pólya for gap series, if ${f_{x,\phi }}$ is entire and has finite order, then it assumes every value infinitely many times. The functions $\phi \in {L^1}(X)$ for which ${g_{x,\phi }}$ is rational a.e. are exactly the finite sums of eigenfunctions of $T$; their poles are all simple and are the inverses of the corresponding eigenvalues. By combining this result with a skew product construction, we can also characterize when ${f_{x,\phi }}$ is rational, provided that $\phi$ takes one of several particular forms.