On the support of ergodic measure for Hénon—Sibony maps on $\mathbb{P}^k$

In this talk, I will briefly introduce the dynamics of one-variable polynomials, the dynamics of Hénon maps on $\mathbb{C}^2$, and then present some basic notions of polynomial automorphisms on $\mathbb{C}^k$. In projective space $\mathbb{P}^k$, Hénon—Sibony maps are the meromorphic extensions of polynomial automorphisms of $\mathbb{C}^k$ under certain conditions. At the end of this talk, we will prove that for a Hénon—Sibony map $f$ on $\mathbb{P}^k$, the support of any ergodic measure with sufficiently large metric entropy is contained in the Julia set.