AI 中文总结
研究次数至少为2且无潜在好约化的多项式，证明具有相同Julia集的多项式动态相关。据此得出任意两个次数至少为2的复多项式，其预周期点集要么重合，要么公共预周期点数量有界，回答了相关猜想并建立了相关结果。
AI 中文摘要
设\(k\)是特征为\(0\)的代数闭完备非阿基米德域。设\(f\)是\(k\)上次数至少为\(2\)且无潜在好约化的多项式。证明若\(g\)是具有相同Julia集的其他多项式，则\(f\)与\(g\)必定动态相关。结果表明，对于任意两个次数至少为\(2\)的复多项式\(f,g\)，要么它们的预周期点集重合，要么它们的公共预周期点数量由仅依赖于次数的常数统一界定，从而回答了DeMarco - Krieger - Ye关于多项式的猜想。还建立了相关结果，可证明DeMarco - Mavraki猜想的特殊情况。
英文摘要
Let $k$ be an algebraically closed, complete non-Archimedean field of residue characteristic $0$. Let $f$ be a polynomial of degree at least $2$ over $k$ which does not have potential good reduction. We prove that if $g$ is any other polynomial with the same Julia set, then $f$ and $g$ must be dynamically related. As a consequence, we show that for any two complex polynomials $f,g$ of degree at least $2$, either their sets of preperiodic points coincide, or the number of their common preperiodic points is uniformly bounded above by a constant depending only on the degrees, thereby answering a conjecture of DeMarco--Krieger--Ye for polynomials. We also establish relative results, allowing us to prove special cases of the DeMarco--Mavraki conjecture.
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