Capacity dimension of the Brjuno set in $\mathbb{C}^n$

TL;DR

The paper addresses the size of the Brjuno-condition complement in $\mathbb{C}^n$ and proves that, for $n\ge 2$, this set has zero $C_\sigma$-capacity when $\sigma>n$ and zero $h_\delta$-Hausdorff measure for $\delta>n+1$. Building on the 1D work of Sadullaev and Rakhimov, the authors develop higher-dimensional preparatory lemmas and construct a divergent lattice-sum potential using measures supported on the hyperplanes given by $k z = p$, to force capacity to vanish. The approach leverages pluripotential-theoretic tools, including $C_\sigma$-capacity, gauge Hausdorff measures, and conformal invariance, to generalize the 1D results to several complex variables. The findings imply that the non-Brjuno set is negligibly small in the capacity and Hausdorff-sense, reinforcing the rigidity of Brjuno-type linearization phenomena in higher dimensions.

Abstract

In this work, we prove that the complement of the Brjuno set in $\mathbb{C}^n$ has zero $C_σ$-capacity with respect to the kernel $k_σ(z,ξ)=\|z-ξ\|^{-2n+2}|\log{\|z-ξ\||^σ}$ for any $σ>n$. In particular, it follows that it has zero $h_δ$-Hausdorff measure with respect to the $h_δ(t)=t^{2n-2}|\log{t}|^{-δ}$, for any $δ>n+1$. This generalizes a previous result of Sadullaev and the second author in dimension one to higher dimensions.

Theorems & Definitions (16)

• Definition 1.1
• Theorem 1.2: Brjuno ABR
• Theorem 1.3: Sadullaev-Rakhimov, AS1
• Theorem 1.4
• Lemma 2.1
• proof
• Definition 3.1
• Remark 3.2
• Lemma 3.3
• proof