Entropy Behavior of Real Rational Maps

Filom, Khashayar

doi:https://doi.org/10.21985/n2-667n-9y25

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Entropy Behavior of Real Rational Maps

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The variation of entropy in a family of dynamical systems is a natural indication of the bifurcations that the family undergoes. In the context of one-dimensional dynamics, Milnor's monotonicity of entropy conjecture (now a theorem of Bruin and van Strien) asserts that for polynomial interval maps with real critical points the entropy level sets (the isentropes) are connected. The literature on the subject is vast and utilizes tools from both real and complex dynamics. In contrast, the entropy behavior of real rational maps is a much less studied albeit much more general setting. In this thesis, we study the continuous real entropy function $h_\Bbb{R}$ that to a map $f\in\Bbb{R}(z)$ of degree $d\geq 2$ assigns the topological entropy of its restriction to the circle $\hat{\Bbb{R}}:=\Bbb{R}\cup\{\infty\}$. The appropriate domain for $h_\Bbb{R}$ is carefully introduced as a certain real subvariety of the dynamical moduli space $\mathcal{M}_d(\Bbb{C})$ of complex rational maps of degree $d$. We continue with proving a rigidity result stating that the function $h_\Bbb{R}$ is locally constant on the intersection of its domain with the structurally-stable component of a rational map in $\mathcal{M}_d(\Bbb{C})$. We then investigate of the monotonicity of $h_\Bbb{R}$ in the case of real quadratic rational maps where $d=2$. Supported by experimental evidence, we show that the level sets of $h_\Bbb{R}$ are connected in certain dynamically defined regions of the moduli space by reducing to the polynomial case while they become disconnected in the region of $(+-+)$-bimodal maps due to a non-polynomial behavior. In stark contrast with the setting of polynomial interval maps, this failure of monotonicity attests to the richer combinatorics of circle maps.

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