Dimension of equilibrium measures for complex maps

2024/02/10 by Ovadia, Snir Ben, He, Yan Mary
#Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2402.07001

Abstract

For certain families of complex maps, we give a formula for the Hausdorff dimension of the equilibrium measure. In particular, given an endomorphism f of \mathbb C\mathbb P^k of algebraic degree d ≥2, and given the equilibrium measure μ with Lyapunov exponents χ₁≥ …≥ χ_k, we show dim_H(μ) = log d∑_{i≤ k}(1)/(χ_i) where dim_H(μ) is the Hausdorff dimension of the measure μ. This gives an answer to the question of Fornæss and Sibony, and proves the Binder-DeMarco Conjecture.

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