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Conformal Geometry and Dynamics

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Real entropy rigidity under quasi-conformal deformations


Author: Khashayar Filom
Journal: Conform. Geom. Dyn. 25 (2021), 1-33
MSC (2020): Primary 37B40, 37E10, 37F31; Secondary 37F10, 37P45
DOI: https://doi.org/10.1090/ecgd/356
Published electronically: March 9, 2021
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Abstract: We set up a real entropy function $h_\Bbb {R}$ on the space $\mathcal {M}’_d$ of Möbius conjugacy classes of real rational maps of degree $d$ by assigning to each class the real entropy of a representative $f\in \Bbb {R}(z)$; namely, the topological entropy of its restriction $f\restriction _{\hat {\Bbb {R}}}$ to the real circle. We prove a rigidity result stating that $h_\Bbb {R}$ is locally constant on the subspace determined by real maps quasi-conformally conjugate to $f$. As examples of this result, we analyze real analytic stable families of hyperbolic and flexible Lattès maps with real coefficients along with numerous families of degree $d$ real maps of real entropy $\log (d)$. The latter discussion moreover entails a complete classification of maps of maximal real entropy.


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Additional Information

Khashayar Filom
Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109
MR Author ID: 1190306
Email: filom@umich.edu

Received by editor(s): May 4, 2019
Received by editor(s) in revised form: December 1, 2020
Published electronically: March 9, 2021
Article copyright: © Copyright 2021 American Mathematical Society