On the measure of scrambled sets of tree maps with zero topological entropy

Abstract

In this paper, we show that if f is a tree map of zero topological entropy and μ is an f-invariant Borel measure then any scrambled set S has zero outer μ-measure (hence μ-measurable). In particular, if S is measurable, it has zero μ-measure.

Keywords:

• tree map
• topological entropy
• division
• Li–Yorke chaos
• scrambled set

2000 Mathematics Subject Classification::

• 37E25
• 37B40

Acknowledgements

I am grateful to the referees for their suggestions that improved the writing of the paper.