Monotonicity of entropy for real quadratic rational maps

Abstract

The monotonicity of entropy is investigated for real quadratic rational maps on the real circle based on the natural partition of the corresponding moduli space into its monotonic, covering, unimodal and bimodal regions. Utilizing the theory of polynomial-like mappings, we prove that the level sets of the real entropy function are connected in the (−+−)-bimodal region and a portion of the unimodal region in . Based on the numerical evidence, we conjecture that the monotonicity holds throughout the unimodal region, but we conjecture that it fails in the region of (+−+)-bimodal maps.