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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Online-Zugang: | https://arxiv.org/abs/2309.13447 |
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| _version_ | 1866908288786890752 |
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| author | Kosek, Marta Stawiska, Malgorzata |
| author_facet | Kosek, Marta Stawiska, Malgorzata |
| contents | We consider a sequence $(p_n)_{n=1}^\infty$ of polynomials with uniformly bounded zeros and $°p_1\geq 1$, $°p_n\geq 2$ for $n\geq 2$, satisfying certain asymptotic conditions. We prove that the function sequence $\left(\frac{1}{°p_n\cdot...\cdot °p_1}\log^+|p_n\circ...\circ p_1|\right)_{n=1}^\infty$ is uniformly convergent in $\mathbb{C}$. The non-autonomous filled Julia set $\mathcal{K}[(p_{n})_{n=1}^\infty]$ generated by the polynomial sequence $(p_{n})_{n=1}^\infty$ is defined and shown to be compact and regular with respect to the Green function. Our toy example is generated by $t_n=\frac{1}{2^{n-1}}T_n,\ n\in\{1,2,...\}$, where $T_n$ is the classical Chebyshev polynomial of degree $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_13447 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Non-autonomous iteration of polynomials in the complex plane Kosek, Marta Stawiska, Malgorzata Complex Variables Dynamical Systems 37F10, 30C15, 30E10, 31A15 We consider a sequence $(p_n)_{n=1}^\infty$ of polynomials with uniformly bounded zeros and $°p_1\geq 1$, $°p_n\geq 2$ for $n\geq 2$, satisfying certain asymptotic conditions. We prove that the function sequence $\left(\frac{1}{°p_n\cdot...\cdot °p_1}\log^+|p_n\circ...\circ p_1|\right)_{n=1}^\infty$ is uniformly convergent in $\mathbb{C}$. The non-autonomous filled Julia set $\mathcal{K}[(p_{n})_{n=1}^\infty]$ generated by the polynomial sequence $(p_{n})_{n=1}^\infty$ is defined and shown to be compact and regular with respect to the Green function. Our toy example is generated by $t_n=\frac{1}{2^{n-1}}T_n,\ n\in\{1,2,...\}$, where $T_n$ is the classical Chebyshev polynomial of degree $n$. |
| title | Non-autonomous iteration of polynomials in the complex plane |
| topic | Complex Variables Dynamical Systems 37F10, 30C15, 30E10, 31A15 |
| url | https://arxiv.org/abs/2309.13447 |