Research Problems in Function Theory — Problem 7.80

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

An inner function is a holomorphic $f:\mathbb D\to\mathbb D$ whose radial boundary values $f^*(e^{i\theta})$ have modulus $1$ almost everywhere. Suppose $f(0)=0$. The boundary map $T_f:\mathbb T\to\mathbb T$ given almost everywhere by $f^*$ preserves normalized length $m=d\theta/(2\pi)$. Its measure-theoretic entropy is \[h_m(T_f)=\sup_{\mathcal P}\lim_{N\to\infty}\frac1N H_m\Big(\bigvee_{j=0}^{N-1}T_f^{-j}\mathcal P\Big), \qquad H_m(\mathcal P)=-\sum_{A\in\mathcal P}m(A)\log m(A), \] where $\mathcal P$ ranges over finite measurable partitions and $0\log0=0$. The Nevanlinna class consists of holomorphic $g$ with $\sup_{0<r<1}\int\log^+|g(re^{i\theta})|\,dm<\infty$, where $\log^+t=\max(0,\log t)$.

Statement recorded in the supplied source.

Determine $h_m(T_f)$. Is it finite if and only if $f'$ belongs to the Nevanlinna class, and, in that case, is \[h_m(T_f)=\int_0^{2\pi}\log|(f')^*(e^{i\theta})|\,\frac{d\theta}{2\pi}? \] The derivative boundary values in this formula are the almost-everywhere radial limits. All inner functions with $f(0)=0$ are included, including rotations; no extra ergodicity assumption is imposed. [S2]

Short English statement

Characterize finite boundary entropy of an inner function fixing zero and test the proposed integral formula in terms of its derivative.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307080 (AMR-022-7080). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), Problem 7.80, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.