We develop a new method, based on pluripotential theory, to study the transfer (Perron-Frobenius) operator induced on Pk=Pk(C) by a holomorphic endomorphism and a suitable continuous weight. This method allows us to prove the existence and uniqueness of the equilibrium state and conformal measure for very general weights (due to Denker-Przytycki-Urbanski in dimension 1 and Urbanski-Zdunik in higher dimensions, both in the case of Holder continuous weights). We establish a number of properties of the equilibrium states, including mixing, K-mixing, mixing of all orders, and an equidistribution of repelling periodic points. Our analytic method replaces all distortion estimates on inverse branches with a unique, global, estimate on dynamical currents, and allows us to reduce the dynamical questions to comparisons between currents and their potentials.

Equilibrium states of endomorphisms of Pk I: Existence and properties

Bianchi F.;
2023-01-01

Abstract

We develop a new method, based on pluripotential theory, to study the transfer (Perron-Frobenius) operator induced on Pk=Pk(C) by a holomorphic endomorphism and a suitable continuous weight. This method allows us to prove the existence and uniqueness of the equilibrium state and conformal measure for very general weights (due to Denker-Przytycki-Urbanski in dimension 1 and Urbanski-Zdunik in higher dimensions, both in the case of Holder continuous weights). We establish a number of properties of the equilibrium states, including mixing, K-mixing, mixing of all orders, and an equidistribution of repelling periodic points. Our analytic method replaces all distortion estimates on inverse branches with a unique, global, estimate on dynamical currents, and allows us to reduce the dynamical questions to comparisons between currents and their potentials.
2023
Bianchi, F.; Dinh, T. -C.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11568/1223409
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