Volume of the hyperbolic cantor sets

摘要

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 ρ ∈ (0,1) there exists a C~1-smooth degree two expanding map f such that the Lebesgue measure of the hyperbolic Cantor set generated by f is ρ.