We consider complex polynomials f(z)=zℓ+c1​ for ℓ∈2N and
c1​∈R, and find some combinatorial types and values of ℓ 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 â„“ sufficiently large and also for a class of
`long-branched' maps of any critical order.Comment: Typos corrected, minor changes, principally to Section