On the support of measures of large entropy for H\'enon-Sibony maps

Let $f$ be a H\'enon-Sibony map of $\mathbb{C}^k$ of algebraic degree $d_+\geq 2$, whose inverse $f^{-1}$ has algebraic degree $d_-$. The topological entropy of $f$ is equal to $\log d_+^{p} = \log d_-^{k-p}$. We show that every ergodic $f$-invariant measure $\nu$ satisfying $h_\nu(f)>\log \max\{ d_+^{p-1},d_-^{k-p-1}\}$ is supported on the Julia set $\mathcal{J}$ of $f$.