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

Let f be a polynomial-like map with dominant topological degree dₜ≥ 2 and let dₖ₋₁<dₜ 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 √dₖ₋₁ dₜ is supported on the Julia set, i.e., the support of the unique measure of maximal entropy μ. The proof is based on the exponential speed of convergence of the measures dₜ⁻ⁿ(fⁿ)^*δₐ towards μ, 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 Pᵏ( C) - a result due to de Th\'elin and Dinh - which does not rely on the existence of a Green current.