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

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