Entropy and periodic points in pseudo-Anosov isotopy classes
Abstract
Let f be a pseudo-Anosov homeomorphism of a closed orientable surface of genus at least two. We prove that, for every C∞ diffeomorphism g isotopic to f, the exponential growth rate of the number of nonempty Nielsen classes of its iterates equals htop(g) if and only if htop(g)=htop(f). Thus Nielsen growth captures the full entropy precisely for representatives minimizing entropy in their isotopy class. For C1+α diffeomorphisms, α>0, we identify this growth rate with the entropy of the global orbit quotient in Fathi's hyperbolic extension and establish Bowen's periodic-point formula for the quotient. We also prove that quotient entropy is upper semicontinuous in the C0 topology throughout the isotopy class and continuous at every C1+α diffeomorphism, even under perturbations by arbitrary homeomorphisms in the class.
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BibTeX
@misc{carrasco2026entropy,
title = {{Entropy and periodic points in pseudo-Anosov isotopy classes}},
author = {Pablo D. Carrasco and Federico Rodriguez-Hertz and Jana Rodriguez-Hertz and Raúl Ures},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26230},
eprint = {2610.26230},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26230},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Entropy and periodic points in pseudo-Anosov isotopy classes
AU - Pablo D. Carrasco
AU - Federico Rodriguez-Hertz
AU - Jana Rodriguez-Hertz
AU - Raúl Ures
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26230
UR - https://arxiv.org/abs/2610.26230
AB - Let f be a pseudo-Anosov homeomorphism of a closed orientable surface of genus at least two. We prove that, for every C^∞ diffeomorphism g isotopic to f, the exponential growth rate of the number of nonempty Nielsen classes of its iterates equals h_top(g) if and only if h_top(g)=h_top(f). Thus Nielsen growth captures the full entropy precisely for representatives minimizing entropy in their isotopy class. For C^1+α diffeomorphisms, α>0, we identify this growth rate with the entropy of the global orbit quotient in Fathi's hyperbolic extension and establish Bowen's periodic-point formula for the quotient. We also prove that quotient entropy is upper semicontinuous in the C^0 topology throughout the isotopy class and continuous at every C^1+α diffeomorphism, even under perturbations by arbitrary homeomorphisms in the class.
ER -