Pressure path metrics on parabolic families of polynomials

Fabrizio Bianchi; Yan Mary He

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arXiv · 2409.10462

Pressure path metrics on parabolic families of polynomials

Fabrizio Bianchi·Yan Mary He

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Abstract

Let $Λ$ be a subfamily of the moduli space of degree $D\ge2$ polynomials defined by a finite number of parabolic relations. Let $Ω$ be a bounded stable component of $Λ$ with the property that all critical points are attracted by either the persistent parabolic cycles or by attracting cycles in $\mathbb C$. We construct a positive semi-definite pressure form on $Ω$ and show that it defines a path metric on $Ω$. This provides a counterpart in complex dynamics of the pressure metric on cusped Hitchin components recently studied by Kao and Bray-Canary-Kao-Martone.

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Fabrizio Bianchi, Yan Mary He. 2024-09-16. Pressure path metrics on parabolic families of polynomials. https://arxiv.org/abs/2409.10462

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