In this paper, we study the problem of control of nonlinear systems over an erasure channel. The stability and performance metric are adopted from the ergodic theory of random dynamical systems to study the almost sure and second moment stabilization problem. The main result of this paper proves that, while there are no limitations for the almost sure stabilization, fundamental limitations arise for the second moment stabilization. In particular, we provide a necessary condition for the second moment stabilization of multi-state single input nonlinear systems expressed in terms of the probability of erasure and positive Lyapunov exponents of the open loop unstable system. The dependence of the limitation result on the Lyapunov exponents highlights, for the first time, the important role played by the global non-equilibrium dynamics of the nonlinear systems in obtaining the performance limitation. This result generalizes the existing results for the stabilization of linear time invariant systems over erasure channels and differs from the existing Bode-like fundamental limitation results for nonlinear systems, which are expressed in terms of the eigenvalues of the linearization.