On the support of measures of large entropy for polynomial-like maps

Let $f$ be a polynomial-like map with dominant topological degree $d_t\geq 2$ and let $d_{k-1}<d_t$ be its dynamical degree of order $k-1$. We show that the support of every ergodic measure whose measure-theoretic entropy is strictly larger than $\log \sqrt{d_{k-1} d_t}$ is supported on the Julia set, i.e., the support of the unique measure of maximal entropy $\mu$. The proof is based on the exponential speed of convergence of the measures $d_t^{-n}(f^n)^*\delta_a$ towards $\mu$, which is valid for a generic point $a$ and with a controlled error bound depending on $a$. Our proof also gives a new proof of the same statement in the setting of endomorphisms of $\mathbb P^k(\mathbb C)$ - a result due to de Th\'elin and Dinh - which does not rely on the existence of a Green current.