arXiv subjects

Nurali Akramov

Publications and source records attributed to Nurali Akramov.

3 recordsLinked to original sources

arXiv2025

Capacity of the α-Brjuno-Rüssmann set

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

math.CV↗

arXiv2025

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

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

math.CV↗

arXiv2025

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

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.

math.CV↗