Entropy and semiconjugacy on surfaces
Abstract
Let g be a C∞ diffeomorphism in the isotopy class of a pseudo-Anosov homeomorphism f such that g and f have the same topological entropy. In 1988, Handel proved that this implies the existence of a semiconjugacy π from g to f. He stated that, in general, there is at least one point x such that π−1(x) is disconnected. We show that this is not the case: for every x, the set π−1(x) is the intersection of a nested sequence of closed topological disks, and hence is connected. We also prove that there is a unique g-invariant probability measure projecting to the measure of maximal entropy of f. This measure is entropy-maximizing, hyperbolic, and Bernoulli, and the semiconjugacy induces a metric isomorphism between the corresponding measure-preserving systems.
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BibTeX
@misc{carrasco2026entropy,
title = {{Entropy and semiconjugacy on surfaces}},
author = {Pablo D. Carrasco and Federico Rodriguez Hertz and Jana Rodriguez Hertz and Raul Ures},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03390},
eprint = {2609.03390},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03390},
}
RIS
TY - UNPB
TI - Entropy and semiconjugacy on surfaces
AU - Pablo D. Carrasco
AU - Federico Rodriguez Hertz
AU - Jana Rodriguez Hertz
AU - Raul Ures
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03390
UR - https://arxiv.org/abs/2609.03390
AB - Let g be a C^∞ diffeomorphism in the isotopy class of a pseudo-Anosov homeomorphism f such that g and f have the same topological entropy. In 1988, Handel proved that this implies the existence of a semiconjugacy π from g to f. He stated that, in general, there is at least one point x such that π^−1(x) is disconnected. We show that this is not the case: for every x, the set π^−1(x) is the intersection of a nested sequence of closed topological disks, and hence is connected. We also prove that there is a unique g-invariant probability measure projecting to the measure of maximal entropy of f. This measure is entropy-maximizing, hyperbolic, and Bernoulli, and the semiconjugacy induces a metric isomorphism between the corresponding measure-preserving systems.
ER -