On the non-monotonicity of entropy for a class of real quadratic rational maps

Abstract

We prove that the entropy function on the moduli space of real quadratic rational maps is not monotonic by exhibiting a continuum of disconnected level sets. This entropy behavior is in stark contrast with the case of polynomial maps, and establishes a conjecture on the failure of monotonicity for bimodal real quadratic rational maps of shape $(+-+)$ which was posed in arXiv:1901.03458 based on experimental evidence.

29 pages, 9 figures. Minor corrections; Corollary 3.10 is changed into a remark; the proof of Lemma 4.1 is shortened. The final version, to appear in the Journal of Modern Dynamics. May vary slightly from the published version