A Riemannian metric on polynomial hyperbolic components
Abstract
We introduce a Riemannian metric on certain hyperbolic components in the moduli space of degree d 2 polynomials. Our metric is constructed by considering the measure-theoretic entropy of a polynomial with respect to some equilibrium state. As applications, we show that the Hausdorff dimension function has no local maximum on such hyperbolic components. We also give a sufficient condition for a point not being a critical point of the Hausdorff dimension function.
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