Snir Ben Ovadia

math.DS — Dynamical Systems

The Binder-DeMarco-Dupont Inequality

Contributed by Snir Ben Ovadia, Aaron Brown, Laura DeMarco

In the setting of endomorphisms of $\mathbb{P}^k(\mathbb{C})$, a dimension formula for the measure of maximal entropy was conjectured by Binder and DeMarco in 2003, asserting that $$\mathrm{dim}_\mathrm{Hausdorff}(\mu)=\log d\sum_{i=1}^k \frac{1}{\chi_i},$$ where $d$ is the degree of the (non-invertible) endomorphism and $\chi_i$'s are the Lyapunov exponents with multiplicity. Indeed, this is well-defined as all Lyapunov exponents are known to be positive due to Briend and Duval. The most significant progress is due to Dupont, who in complex-dimension 2, proved the dimension lower-bound holds; And recently by Genes, who proved the equality for two-dimensional polynomial skew-product maps which satisfy a Lyapunov exponent condition. Recently, He and Bianchi were able to find a family of simple counter-examples. In Appendix A we present a single ``hands on'' description of such a counter example, which is given by a quadratic polynomial skew-product. In this manuscript, we provide the proof, which was achieved by lightly prompting ChatGPT, of extending Dupont’s inequality to all complex dimensions. We wish to share this manuscript with the community as we believe that this provides a full picture of the recently settled Binder-DeMarco conjecture. We do not claim ownership of the proof.