A Proof of the Marmi-Moussa-Yoccoz conjecture for rotation numbers of high type

TL;DR

This paper proves the Marmi-Moussa-Yoccoz conjecture that a specific error function related to Siegel disk size is Hölder continuous with exponent 1/2 for high type rotation numbers, using renormalization techniques.

Contribution

It establishes the conjecture for high type numbers by applying the Inou-Shishikura renormalization framework, advancing understanding of Siegel disk approximation.

Findings

Upsilon function is Hölder continuous with exponent 1/2 for high type numbers

Uses renormalization invariants to prove regularity of the error function

Confirms conjecture within a specific class of rotation numbers

Abstract

Marmi Moussa and Yoccoz conjectured that some error function Upsilon, related to the approximation of the size of Siegel disk by some arithmetic function of the rotation number theta, is a Holder continuous function of theta with exponent 1/2. Using the renormalization invariant class of Inou and Shishikura, we prove this conjecture for the restriction of Upsilon to a class of high type numbers.