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 $(n_k{)}_{k⩾1}$ such that $({\sigma }_k{)}_{k⩾1}:=(n_{k+1}-\alpha n_k-\beta \ln n_k{)}_{k⩾1}$ 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 $(n_k{)}_{k⩾1}$ such that $({\sigma }_k{)}_{k⩾1}$ 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 $(n_k{)}_{k⩾1}$ such that $({\sigma }_k{)}_{k⩾1}$ 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.