On bounded distortions of maps in the line

Abstract

Let and be disjoint closed subintervals of [0,1] and let be a map such that is a diffeomorphism onto [0,1] and 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 maps. In this note, we give an example that shows that SBD is indeed stronger than BD.

Keywords:

• bounded distortion
• scaling function
• Cantor sets

AMS Subject Classifications::

• 26A18
• 37E05

Acknowledgements

We thank the anonymous referee for her/his valuable comments and suggestions. I. Garcia is partially supported by CAI+D2009 No. 62-310. C.G. Moreira is partially supported by CNPq and by the Balzan Research Project of J. Palis.