Equilibrium measures on saddle sets of holomorphic maps on \mathbb{P }^2 |
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Authors: | John Erik Fornaess Eugen Mihailescu |
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Institution: | 1. Institutt for Matematiske Fag, Sentralbygg 2, Alfred Getz Vei 1, 7491, Trondheim, Norway 2. Mathematical Sciences Center, Tsinghua University, 131 Jin Chun Yuan West building, Haidian District, Beijing, ?100084, China 3. Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700, Bucharest, Romania
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Abstract: | We consider the case of hyperbolic basic sets $\Lambda $ of saddle type for holomorphic maps $f:{\mathbb{P }}^2{\mathbb{C }}\rightarrow {\mathbb{P }}^2{\mathbb{C }}$ . We study equilibrium measures $\mu _\phi $ associated to a class of Hölder potentials $\phi $ on $\Lambda $ , and find the measures $\mu _\phi $ of iterates of arbitrary Bowen balls. Estimates for the pointwise dimension $\delta _{\mu _\phi }$ of $\mu _\phi $ that involve Lyapunov exponents and a correction term are found, and also a formula for the Hausdorff dimension of $\mu _\phi $ in the case when the preimage counting function is constant on $\Lambda $ . For terminal/minimal saddle sets we prove that an invariant measure $\nu $ obtained as a wedge product of two positive closed currents, is in fact the measure of maximal entropy for the restriction $f|_\Lambda $ . This allows then to obtain formulas for the measure $\nu $ of arbitrary balls, and to give a formula for the pointwise dimension and the Hausdorff dimension of $\nu $ . |
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