Journal article
Pablo D. Carrasco, Jana Rodriguez-Hertz, Federico Rodriguez-Hertz, Raúl Ures
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APA
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Carrasco, P. D., Rodriguez-Hertz, J., Rodriguez-Hertz, F., & Ures, R. Entropy and periodic points in pseudo-Anosov isotopy classes.
Chicago/Turabian
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Carrasco, Pablo D., Jana Rodriguez-Hertz, Federico Rodriguez-Hertz, and Raúl Ures. “Entropy and Periodic Points in Pseudo-Anosov Isotopy Classes” (n.d.).
MLA
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Carrasco, Pablo D., et al. Entropy and Periodic Points in Pseudo-Anosov Isotopy Classes.
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@article{pablo-a,
title = {Entropy and periodic points in pseudo-Anosov isotopy classes},
author = {Carrasco, Pablo D. and Rodriguez-Hertz, Jana and Rodriguez-Hertz, Federico and Ures, Raúl}
}
Let $f$ be a pseudo-Anosov homeomorphism of a closed orientable surface of genus at least two. We prove that, for every $C^\infty$ diffeomorphism $g$ isotopic to $f$, the exponential growth rate of the number of nonempty Nielsen classes of its iterates equals $h_{\mathrm{top}}(g)$ if and only if
$h_{\mathrm{top}}(g)=h_{\mathrm{top}}(f)$. Thus Nielsen growth captures the full entropy precisely for representatives minimizing entropy in their isotopy class. For $C^{1+\alpha}$ diffeomorphisms, $\alpha>0$, 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 $C^{1+\alpha}$ diffeomorphism, even under perturbations by arbitrary homeomorphisms in the class.