Distribution of bounded real sequences and a question of Astorg and Boc Thaler
Zhangchi Chen, Zihao Ye, Weizhe Zheng
TL;DR
This work links the dynamics of skew-products tangent to the identity in $\mathbb{C}^2$ to number-theoretic distribution questions, addressing when the phase sequence $\sigma_k=n_{k+1}-\alpha n_k-\beta\ln n_k$ converges or converges to a cycle. For algebraic $\alpha>1$ (Pisot), it establishes sharp necessary and sufficient conditions on $\theta=\frac{\beta\ln\alpha}{\alpha-1}$: convergence requires $\theta\in\frac{1}{P(1)}\mathbb{Z}$, while convergence to a cycle of period $\ell$ requires $\theta\in\frac{1}{\ell P(1)}\mathbb{Z}$. The authors construct explicit sequences via complete homogeneous polynomials and Weyl equidistribution to realize these convergences, yielding explicit new wandering-domain examples; they also extend distribution modulo $1$ results for linear recurrences, generalizing Dubickas' theorems and providing a broader toolkit for analyzing phase dynamics in complex systems.
Abstract
Astorg and Boc Thaler studied the dynamics of certain skew-product $f(z,w)=(p(z),q(z,w))$ tangent to the identity, 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: the condition that $θ\in\mathbb{Q}$ is necessary and sufficient for the convergence of $(σ_k)_{k\geqslant 1}$ to a cycle. 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 convergence of $(σ_k)_{k\geqslant 1}$. Combined with the work of Astorg and Boc Thaler, our result provides explicit new examples of polynomial skew-products in $\mathbb{C}^2$ with wandering domains of rank one. We also prove related results on the distribution modulo one of linear recurrent sequences, generalizing theorems of Dubickas.