A Dynamical Fekete-Szegő Theorem
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arXiv
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| Natura: | Preprint |
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2026
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| author | Bayraktar, Turgay Efe, Melike |
| author_facet | Bayraktar, Turgay Efe, Melike |
| contents | Let $E\subset\Bbb{C}$ be a compact set symmetric with respect to the real axis. A classical theorem of Fekete-Szegő asserts that such a compact set is of logarithmic capacity at least one if and only if it admits approximation by algebraic integers whose Galois conjugates lie arbitrarily close to $E$. In this note we prove a dynamical analogue of this phenomenon. When $\mathrm{cap}(E)=1$, we also show that the algebraic polynomials arising from the Fekete-Szegő theorem generate filled Julia sets $K_{P_n}$ which converge to the polynomially convex hull $Pc(E)$ in the Klimek topology, while their Brolin measures converge to the equilibrium measure $μ_E$. In particular, when $E\subset\Bbb{R}$, this provides a genuine approximation of $E$ by algebraic filled Julia sets.
As an arithmetic application, we prove that the Rumely height associated to $E$ arises as a limit of canonical dynamical heights in the sense of Call and Silverman, giving a dynamical counterpart to the equidistribution theorems of Bilu and Rumely. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_01684 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Dynamical Fekete-Szegő Theorem Bayraktar, Turgay Efe, Melike Dynamical Systems Complex Variables Number Theory 37F50, 11G50, 31A15 Let $E\subset\Bbb{C}$ be a compact set symmetric with respect to the real axis. A classical theorem of Fekete-Szegő asserts that such a compact set is of logarithmic capacity at least one if and only if it admits approximation by algebraic integers whose Galois conjugates lie arbitrarily close to $E$. In this note we prove a dynamical analogue of this phenomenon. When $\mathrm{cap}(E)=1$, we also show that the algebraic polynomials arising from the Fekete-Szegő theorem generate filled Julia sets $K_{P_n}$ which converge to the polynomially convex hull $Pc(E)$ in the Klimek topology, while their Brolin measures converge to the equilibrium measure $μ_E$. In particular, when $E\subset\Bbb{R}$, this provides a genuine approximation of $E$ by algebraic filled Julia sets. As an arithmetic application, we prove that the Rumely height associated to $E$ arises as a limit of canonical dynamical heights in the sense of Call and Silverman, giving a dynamical counterpart to the equidistribution theorems of Bilu and Rumely. |
| title | A Dynamical Fekete-Szegő Theorem |
| topic | Dynamical Systems Complex Variables Number Theory 37F50, 11G50, 31A15 |
| url | https://arxiv.org/abs/2603.01684 |