历史依赖的不稳定方向与Binder--DeMarco猜想

中文摘要 AI 辅助

我们建立了满足适当扩张和支配条件的$\mathbb P^2=\mathbb P^2(\mathbb C)$上次数为$d$的全纯自同态的平衡测度的维数界和精确维数公式。记其Lyapunov指数为$\lambda_1>\lambda_2>0$。如果两条终止于同一点的不同逆历史确定了不同的快方向，我们证明：当$\lambda_2>\log d$时，${\mathrm dim}_H \mu_F>\log d/\lambda_1+\log d/\lambda_2$；当$\lambda_2\ge2\log d$时，${\mathrm dim}_H \mu_F=2\log d/\lambda_2$。将这些结果应用于一个显式的二次族，我们在一个非空开集的全纯自同态上否证了Binder--DeMarco猜想。证明将Li--Pan--Tong--Xu的投影增长策略适应到全纯情形，结合了强叶几何和用于自同态的Ledrappier--Young理论，以及Wu的受限和估计的多维推广。

英文摘要

We establish dimension bounds and an exact dimension formula for equilibrium measures of degree $d$ holomorphic endomorphisms of $\mathbb P^2=\mathbb P^2(\mathbb C)$ satisfying suitable expansion and domination conditions. Write $λ_1>λ_2>0$ for their Lyapunov exponents. If two inverse histories ending at the same point determine different fast directions, we prove that ${\mathrm dim}_H μ_F>\log d/λ_1+\log d/λ_2$ when $λ_2>\log d$, and that ${\mathrm dim}_H μ_F=2\log d/λ_2$ when $λ_2\ge2\log d$. Applying these results to an explicit quadratic family, we disprove the Binder--DeMarco conjecture on a non-empty open set of holomorphic endomorphisms. The proof adapts the projection growth strategy of Li--Pan--Tong--Xu to the holomorphic setting, combining strong leaf geometry and the Ledrappier--Young theory for endomorphisms with a multidimensional extension of Wu's restricted sum estimate.

发表机构

• Università di Pisa（比萨大学）
• University of Oklahoma（俄克拉荷马大学）

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