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

Abstract

In this work, we prove that the complement of the Brjuno set $\mathcal{B}$ has a zero capacity with respect to the kernel $k_{\sigma }^1(z,\xi )={\ln }^2|z-\xi |{\left|\ln \ln \left(e+\frac{1}{|z-\xi |}\right)\right|}^{\sigma }$ for any $\sigma >2$. Similarly, the complement of the Perez-Marco set $\mathcal{P}\mathcal{M}$ has a zero capacity with respect to the kernel $k_{\sigma }^2(z,\xi )={\ln }^2\ln \left(e+\frac{1}{|z-\xi |}\right)\cdot {\ln }^{\sigma }\ln \ln \left(e^3+\frac{1}{|z-\xi |}\right)$ for any $\sigma >2$.