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

Let f be a polynomial-like map with dominant topological degree \(d_t\ge 2\) and let \(d_{k-1}<d_t\) be its dynamical degree of order \(k-1\). We show that 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 Thélin and Dinh – which does not rely on the existence of a Green current.