  We consider complex polynomials $f(z) = z^\ell+c_1$ for $\ell \in 2\N$ and
$c_1 \in \R$, 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.
