On a question of Astorg and Boc Thaler
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2025
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| _version_ | 1866910137322569728 |
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| author | Chen, Zhangchi Ye, Zihao Zheng, Weizhe |
| author_facet | Chen, Zhangchi Ye, Zihao Zheng, Weizhe |
| contents | Astorg and Boc Thaler studied the dynamics of certain skew-product tangent to the identity on $\mathbb{C}^2$, with two real parameters $α>1$ and $β$ derived from its coefficients. They proved that if there exists an increasing sequence of positive integers $(n_k)_{k\geqslant 1}$ such that $(σ_k)_{k\geqslant 1}:=(n_{k+1}-αn_k-β\ln n_k)_{k\geqslant 1}$ converges, then $f$ admits wandering domains of rank one. They also proved that for $α>1$ with the Pisot property, the condition that $θ:=\frac{β\lnα}{α-1}$ is rational is sufficient for the existence of $(n_k)_{k\geqslant 1}$ such that $(σ_k)_{k\geqslant 1}$ converges to a cycle. They asked if this condition is necessary.
When $α$ 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~$α$, we prove that $θ\in\frac{1}{P(1)}\mathbb{Z}$ is necessary and sufficient for the existence of $(n_k)_{k\geqslant 1}$ such that $(σ_k)_{k\geqslant 1}$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_21324 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a question of Astorg and Boc Thaler Chen, Zhangchi Ye, Zihao Zheng, Weizhe Dynamical Systems Number Theory Primary 37F10, Secondary 11J71, 11K16, 37F80 Astorg and Boc Thaler studied the dynamics of certain skew-product tangent to the identity on $\mathbb{C}^2$, with two real parameters $α>1$ and $β$ derived from its coefficients. They proved that if there exists an increasing sequence of positive integers $(n_k)_{k\geqslant 1}$ such that $(σ_k)_{k\geqslant 1}:=(n_{k+1}-αn_k-β\ln n_k)_{k\geqslant 1}$ converges, then $f$ admits wandering domains of rank one. They also proved that for $α>1$ with the Pisot property, the condition that $θ:=\frac{β\lnα}{α-1}$ is rational is sufficient for the existence of $(n_k)_{k\geqslant 1}$ such that $(σ_k)_{k\geqslant 1}$ converges to a cycle. They asked if this condition is necessary. When $α$ 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~$α$, we prove that $θ\in\frac{1}{P(1)}\mathbb{Z}$ is necessary and sufficient for the existence of $(n_k)_{k\geqslant 1}$ such that $(σ_k)_{k\geqslant 1}$ 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. |
| title | On a question of Astorg and Boc Thaler |
| topic | Dynamical Systems Number Theory Primary 37F10, Secondary 11J71, 11K16, 37F80 |
| url | https://arxiv.org/abs/2511.21324 |