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arXiv · 1901.03458

Monotonicity of entropy for real quadratic rational maps

Khashayar Filom
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Abstract

The monotonicity of entropy is investigated for real quadratic rational maps on the real circle $\mathbb{R}\cup\{\infty\}$ based on the natural partition of the corresponding moduli space $\mathcal{M}_2(\mathbb{R})$ 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 $h_\mathbb{R}$ are connected in the $(-+-)$-bimodal region and a portion of the unimodal region in $\mathcal{M}_2(\mathbb{R})$. 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.

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Khashayar Filom. 2019-01-11. Monotonicity of entropy for real quadratic rational maps. https://doi.org/10.1088/1361-6544/ac15aa

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