Abstract
Marmi, Moussa and Yoccoz conjectured that some error function \(\Upsilon \), related to the approximation of the size of Siegel disks by some arithmetic function of the rotation number \(\theta \), is a Hölder 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.
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Notes
We could have taken \(a_n\) to be the floor of \(1/\alpha _n\) instead of the nearest integer. There is a simple way to pass from one convention to the other, since \((a_n,s_n)\) in one convention depends solely on \((a_n,s_n)\) in the other. The important thing is to have a way to label the intervals on which the maps \(H_n\) defined below are bijections.
It can also be proved using an inductive computation on \(H_n\) as a Möbius map on fundamental intervals.
We will thus work with two topologies on \(\mathcal {I}\mathcal {S}_0\), the one induced by \(\mathrm{d_{\mathrm{Teich}} }\), for which it is complete but not pre-compact, and the one from compact-open topology, for which it is pre-compact but not complete.
The class \(\mathcal {I}\mathcal {S}_0\) is denoted by \(\mathcal {F}_1\) in [16]. All above statements, except the existence of uniform \(\hat{\mathbf {k}}\) and \(\mathbf {k}\), follow from Theorem 2.1, and Main Theorems 1, 3 in [16]. The existence of uniform constants also follows from those results but requires some extra work. A detailed treatment of these statements is given in [12, Proposition 1.4] and [7, Proposition 12].
The operator \(\mathrm{Dom }\) denotes “the domain of definition” (of a map).
The sets \({\mathcal {C}}_h^{-k}\) and \((\mathcal {C}^\sharp _h)^{-k}\) defined here are (strictly) contained in the sets denoted by \(V^{-k}\) and \(W^{-k}\) in [7]. The set \(\Phi _h({\mathcal {C}}_h^{-k}\cup (\mathcal {C}^\sharp _h)^{-k})\) is contained in the union
$$\begin{aligned} D^\sharp _{-k} \cup D_{-k} \cup D''_{-k} \cup D'_{-k+1} \cup D_{-k+1} \cup D^\sharp _{-k+1} \end{aligned}$$in the notations used in [16, Section 5.A].
The difficulty here is that we do not know a priori that the image of \({\widetilde{\Delta }}(R(f))\) under \(\Phi _f^{-1}\) is equal to \(\Delta (f)\cap {\mathcal {P}}_f\), nor in fact any of the two inclusions. Another is that \(\Delta (f)\cap {\mathcal {P}}_f\) and \({\widetilde{\Delta }}({\mathcal {R}}(f)) \cap \mathrm{Dom }(\Phi _f^{-1})\) may be disconnected. Last, it is not known whether \(\Delta (f)\) may have a non locally connected boundary, although it is unlikely: it is for instance conjectured that the boundary of all polynomial Siegel disks are Jordan curves.
Indeed, we can improve the estimates to the second order, involving \(|z^2|\) and \(\mathrm{d_{\mathrm{Teich}} }^2\) here, but since the coefficients of the first terms are non-zero, we do not care about the higher order terms.
We wish to compare \(L_\beta \) to a model map near the right end of the domain, but, the issue here is that near \(1/\beta \) we do not a priori have a well behaving normalization for a model map, like \(H_\beta (0)=cp _\beta \).
The subscript \(T\) stands for Teichmüller, but our proof is valid for any metric with the required properties stated here.
For maps in \(\mathcal {I}\mathcal {S}_{N}\), we cannot use this trick because there is also the term \(d_T(f_0,g_0)\) to take into account, and \(d_T((s f s^{-1})_0,g_0) = d_T(sf_0 s^{-1},g_0)\) could be a lot bigger than \(d_T(f_0,g_0)\), even when \(\alpha (f)\) and \(\alpha (g)\) are small.
According to discussion with them, the number \(N\) provided by Inou and Shishikura is likely to be no less than \(20\).
This permutation is done only at this point of the article, so it is safe.
This estimate is sufficient but is far from optimal. A bound of order \(\beta _{n_1-2}^2(\alpha _{n_1-1}+\alpha '_{n_1-1})\) can be obtained.
The proof could be as well carried out without this improvement.
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The first author would like to thank the Leverhulme Trust in London for their partial financial support while carrying out this research.
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Cheraghi, D., Chéritat, A. A proof of the Marmi–Moussa–Yoccoz conjecture for rotation numbers of high type. Invent. math. 202, 677–742 (2015). https://doi.org/10.1007/s00222-014-0576-2
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DOI: https://doi.org/10.1007/s00222-014-0576-2