Dendrites and measures with discrete spectrum

TL;DR

This paper characterizes dendrites where all zero-entropy invariant measures have discrete spectrum, proving this property holds when the endpoint set's closure is countable, thus nearly completing the classification of such dendrites.

Contribution

It proves that dendrites with a countable closure of their endpoint set have all zero-entropy invariant measures with discrete spectrum, solving an open problem.

Findings

Dendrites with countable endpoint set closure have all zero-entropy invariant measures with discrete spectrum.

The result nearly completes the classification of dendrites with this spectral property.

Abstract

We are interested in dendrites for which all invariant measures of zero-entropy mappings have discrete spectrum, and we prove that this holds when the closure of the endpoint set of the dendrite is countable. This solves an open question which was around for awhile, almost completing the characterization of dendrites with this property.