On the capacity dimensions of the Brjuno and Perez-Marco sets

2025/07/21 by Nurali Akramov, Akramov, Nurali, Abduvahhob Ashirov +1
Economics, Econometrics and Finance · #Complex Variables (math.CV) #Economic theories and models #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2507.15980

openalex publication_date 2025/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this work, we prove that the complement of the Brjuno set B has a zero capacity with respect to the kernel k¹_σ(z,ξ)=ln²|z-ξ||lnln(e+(1)/(|z-ξ|))|^σ for any σ> 2. Similarly, the complement of the Perez-Marco set PM has a zero capacity with respect to the kernel k²_σ(z,ξ) = ln²ln(e+\frac1 | z - ξ|) ⋅ln^σlnln(e³+\frac1 | z - ξ|) for any σ>2.

Citations

• Dynamics in one complex variable: introductory lectures

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