On a question of Astorg and Boc Thaler

Astorg and Boc Thaler studied the dynamics of certain skew-product tangent to the identity on $\mathbb{C}^2$, with two real parameters $\alpha>1$ and $\beta$ derived from its coefficients. They proved that if there exists an increasing sequence of positive integers $(n_k)_{k\geqslant 1}$ such that $(\sigma_k)_{k\geqslant 1}:=(n_{k+1}-\alpha n_k-\beta\ln n_k)_{k\geqslant 1}$ converges, then $f$ admits wandering domains of rank one. They also proved that for $\alpha>1$ with the Pisot property, the condition that $\theta:=\frac{\beta\ln\alpha}{\alpha-1}$ is rational is sufficient for the existence of $(n_k)_{k\geqslant 1}$ such that $(\sigma_k)_{k\geqslant 1}$ converges to a cycle. They asked if this condition is necessary. When $\alpha$ is an algebraic number, we answer the question of Astorg and Boc Thaler in the affirmative. Furthermore, denoting by $P(x)\in\mathbb{Z}[x]$ the minimal polynomial of~$\alpha$, we prove that $\theta\in\frac{1}{P(1)}\mathbb{Z}$ is necessary and sufficient for the existence of $(n_k)_{k\geqslant 1}$ such that $(\sigma_k)_{k\geqslant 1}$ converges. Combined with the work of Astorg and Boc Thaler, our result provides explicit new examples of skew-products on $\mathbb{C}^2$ with wandering domains of rank one.