We show that the growth inequality rate $$\limsup \frac{1}{n} \log (\# Fix (f^n))\geq \log d$$ holds for branched coverings of degree $d$ of the sphere $S^2$ having a completely invariant simply connected region $R$ with locally connected boundary, except in some degenerate cases with known couterexamples.

中文翻译：

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球体分支覆盖物和增长率不平等

我们证明了增长不平等率 $$\limsup \frac{1}{n} \log (\# Fix (f^n))\geq \log d$$ 对球的 $d$ 度的分支覆盖成立$S^2$ 具有完全不变的简单连接区域 $R$ 与局部连接边界，除了在一些已知的 couterexamples 的退化情况下。