Localization of collapsed positive entropy

Determine whether, for a smooth surface diffeomorphism isotopic to a pseudo-Anosov homeomorphism and semiconjugate to that pseudo-Anosov representative under equality of topological entropies, every positive entropy collapsed by the semiconjugacy is localized entirely on periodic orbits.

Background

The paper studies a smooth surface diffeomorphism that is isotopic to a pseudo-Anosov map and has the same topological entropy as that representative. Handel’s semiconjugacy provides a factor map to the pseudo-Anosov dynamics. The paper proves that all fibers of this semiconjugacy are cellular and that, for the distinguished measure of maximal entropy, the semiconjugacy is a metric isomorphism. The authors then state a broader conjectural claim concerning invariant measures whose dynamics are collapsed by the factor map: any entropy that is lost under the semiconjugacy should occur only on periodic orbits. The paper does not establish this claim.