Distributions of bounded real sequences and Astorg-Boc Thaler's question

2025/11/26 by Chen, Zhangchi, Ye, Zihao
#11J71 #11K16 #37F10 #37F40 #37F80 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2511.21324

Abstract

While studying the existence of wandering domains in complex dynamics, Astorg-Boc Thaler posed a question concerning the distribution of bounded real sequences: for α>1 that has the Pisot property, is (βlnα)/(α-1) being rational necessary for the existence of an increasing sequence of integers (n_k)_{k\geqslant 1} such that (n_k+1-αn_k-βln n_k)_{k\geqslant 1} converges to a cycle? When α is an algebraic number, we answer Astorg-Boc Thaler's question in the affirmative: the condition θ being rational is necessary and sufficient for the convergence to a cycle. Furthermore, let P(x)∈ℤ[x] be the minimal polynomial of α. We prove that P(1)θ∈ℤ is a necessary condition, while (P(1)θ)/(gcd(P(1),P'(1)))∈ℤ is a sufficient condition, for the convergence to a cycle of length 1. As an application, we present explicit new examples of polynomial skew products in ℂ² with wandering domains.

Citations

• Dynamics of skew-products tangent to the identity
• A two-dimensional polynomial mapping with a wandering Fatou component
• On a Problem in Diophantine Approximation

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