Dynamics on a Siegel-disk boundary

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.

A Siegel disk $S$ for a holomorphic map $f$ is an invariant domain on which $f$ is holomorphically conjugate to an irrational rotation of the unit disk. Consider its boundary dynamics wherever, as in the source's holomorphic setting, $f$ extends to a neighbourhood of $\overline S$. A periodic boundary point satisfies $f^p(x)=x$ for some $p\geq1$. The lower Lyapunov exponent at $x$ is $\underline\chi(x)=\liminf_{n\to\infty}n^{-1}\log\|(f^n)'(x)\|$, with derivative norm in a smooth spherical metric; positivity is independent of a smooth change of metric on the compact invariant boundary. For a continuous map $T$ on a compact metric space, its topological entropy is $h_{\rm top}(T)=\lim_{\varepsilon\downarrow0}\limsup_{n\to\infty}n^{-1}\log s_n(\varepsilon)$, where $s_n(\varepsilon)$ is the largest size of a set whose distinct points are separated by more than $\varepsilon$ in $\max_{0\leq j<n}d(T^jx,T^jy)$.

Statement recorded in the supplied source.

Can $\partial S$ contain periodic points? Can it contain points of positive Lyapunov exponent? Is $h_{\rm top}(f|_{\partial S})=0$ always true? [S2]

Short English statement

Can a Siegel-disk boundary contain periodic points or points with positive derivative growth, and must its boundary dynamics always have zero entropy?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300051 (AMR-052-0051). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Feliks Przytycki, "On Invariant Measures for Iterations of Holomorphic Maps", in Ben Bielefeld and Mikhail Lyubich (eds.), Problems in Holomorphic Dynamics (1992), Problem 1.4, pp. 30; arXiv:math/9205209v1. https://arxiv.org/pdf/math/9205209.