ABSTRACT In this paper, we study hyperbolic Cantor sets on the line. We prove that the Lebesgue measure of a hyperbolic Cantor set generated by a degree two expanding map satisfying a certain Zygmund condition is zero. Moreover, we show that for any there exists a -smooth degree two expanding map f such that the Lebesgue measure of the hyperbolic Cantor set generated by f is ρ.

中文翻译：

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双曲康托集的体积

摘要 在本文中，我们研究了线上的双曲康托集。我们证明了由满足某个齐格蒙德条件的二阶扩展映射生成的双曲康托集的勒贝格测度为零。此外，我们证明，对于任何存在 - 平滑度二扩展映射 f，使得由 f 生成的双曲康托集的 Lebesgue 测度是 ρ。