On the support of measures of large entropy for Hénon-Sibony maps

Let $f$ be a Hénon-Sibony map of ${\mathbb{C}}^k$ of algebraic degree $d_{+}\ge 2$, 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 $f$.