## S. Bazarbaev - F. Bianchi - K. Rakhimov

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

created by bianchi on 23 Sep 2024

[

BibTeX]

*preprint*

**Inserted:** 23 sep 2024

**Last Updated:** 23 sep 2024

**Year:** 2024

**Abstract:**

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.

**Tags:**
PRIN2022-MFDS