On bounded distortions of maps in the line

摘要

Let I_1 and I_2 be disjoint closed subintervals of [0,1] and let F: I_1 ∪ I_2 → [0, 1] be a C~1 map such that F|_li is a diffeomorphism onto [0,1] and F' > 1 on its domain. The repeller K of F is a Cantor set. If F satisfies the bounded distortion property (BD), then K has Hausdorff dimension t smaller than 1 and its t-dimensional Hausdorff measure is finite and positive. Another related property is the strong bounded distortion (SBD), which is needed to define the scaling function of repellers associated with C~(1+) maps. In this note, we give an example that shows that SBD is indeed stronger than BD.