AI 中文总结
本文研究整函数族$f_\lambda(z)=f(\lambda z)$的Julia集中非逃逸点集合的Hausdorff维数，给出当$|\lambda|\to 0$时维数趋于1的条件，并推广了指数函数情形。
AI 中文摘要
对于整函数$f$和非零复数$\lambda$，设$f_\lambda(z)=f(\lambda z)$。我们给出$f$的条件，这些条件蕴含当$|\lambda|\to 0$时，$f_\lambda$的Julia集中非逃逸点集合的Hausdorff维数趋于$1$。事实上，我们给出了该维数关于$\lambda$的上界。这推广了先前关于$f(z)=\exp z$情形的结果。
英文摘要
For an entire function $f$ and a non-zero complex number $λ$, let $f_λ(z)=f(λz)$. We give conditions on $f$ which imply that the Hausdorff dimension of the set of non-escaping points in the Julia set of $f_λ$ tends to $1$ as $|λ|\to 0$. In fact, we give an upper bound for this dimension in terms of $λ$. This generalizes earlier results concerned with the case that $f(z)=\exp z$.
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