Monotonicity of entropy for real quadratic rational maps

Abstract

The monotonicity of entropy is investigated for real quadratic rational maps on the real circle R∪{∞} based on the natural partition of the corresponding moduli space M2(R) 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 hR are connected in the (−+−)-bimodal region and a portion of the unimodal region in M2(R) . 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.