Let $f : V \to U$ be a (generalized) polynomial-like map. Suppose thatharmonic measure $\omega= \omega(\cdot,\infty)$ on the Julia set $J_{f}$ is equal tothe measure of maximal entropy $m$ for $f : J_{f} \hookleftarrow$. Then thedynamics $(f,V,U)$ is called maximal. We are going to give a criterion for thedynamics to be conformally equivalent to a maximal one, that is to beconformally maximal. In the second part of this paper we construct aninvariant ‘harmonic’ measure $\mu$ such that ${d\mu}/{d\omega}$ is Hölderfor certain dynamics. This allows us to prove in this class of dynamicalsystems that $\omega\approx m$ is necessary and sufficient for $(f,V,U)$ to beconformally maximal. In the particular case when $f$ is expanding and $J_{f}$is a circle, our result becomes a theorem of Shub and Sullivan; so throughoutthe paper we are dealing with an analog of a theorem of Shub and Sullivan on‘wild’ (e.g. totally disconnected) $J_{f}$ and for certain non-expanding$f$. We also construct (under certain assumptions) invariant harmonic measure on$J_{f}$. In this respect, our work stems from one of the works of Carleson.