For holomorphic maps tangent to the identity at a fixed point, Hakim studied the local dynamics in the so-called non-degenerate characteristic directions, showing the existence of attracting petals and domains under certain conditions. Abate showed in dimension two that the attracting petals always exist. However the question remains whether there always exist attracting petals or domains in degenerate characteristic directions. Molino showed the existence of attracting petals when the so-called residual index in a characteristic direction is non-zero (under some mild assumptions). In this talk, we give several examples showing the existence (or non-existence) of attracting domains in degenerate characteristic directions with zero residual index. This is joint work with Berit Stensones.