Complex maps without invariant densities

Abstract

We consider complex polynomials $f(z)=z^{\ell }+c_1$ for $\ell \in 2\mathbb{N}$ and $c_1\in \mathbb{R}$, and find some combinatorial types and values of $\ell$ such that there is no invariant probability measure equivalent to conformal measure on the Julia set. This holds for particular Fibonacci-like and Feigenbaum combinatorial types when $\ell$ sufficiently large and also for a class of `long-branched' maps of any critical order.

Typos corrected, minor changes, principally to Section 6