Dimension of equilibrium measures for complex maps

Snir Ben Ovadia; Yan Mary He

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arXiv · 2402.07001

Dimension of equilibrium measures for complex maps

Snir Ben Ovadia·Yan Mary He

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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 $d \ge2$, and given the equilibrium measure $\mu$ with Lyapunov exponents $\chi_1\geq \ldots\geq \chi_k$, we show $\dim_\mathrm{H}(\mu) = \log d\sum_{i\leq k}\frac{1}{\chi_i}$ where $\dim_\mathrm{H}(\mu)$ is the Hausdorff dimension of the measure $\mu$. This gives an answer to the question of Forn{\ae}ss and Sibony, and proves the Binder-DeMarco Conjecture.

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Snir Ben Ovadia, Yan Mary He. 2024-02-10. Dimension of equilibrium measures for complex maps. https://arxiv.org/abs/2402.07001

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