  We derive and solve renormalization group equations that allow for the
resummation of subleading power rapidity logarithms. Our equations involve
operator mixing into a new class of operators, which we term the "rapidity
identity operators", that will generically appear at subleading power in
problems involving both rapidity and virtuality scales. To illustrate our
formalism, we analytically solve these equations to resum the power suppressed
logarithms appearing in the back-to-back (double light cone) limit of the
Energy-Energy Correlator (EEC) in $\mathcal{N}$=4 super-Yang-Mills. These
logarithms can also be extracted to $\mathcal{O}(\alpha_s^3)$ from a recent
perturbative calculation, and we find perfect agreement to this order. Instead
of the standard Sudakov exponential, our resummed result for the subleading
power logarithms is expressed in terms of Dawson's integral, with an argument
related to the cusp anomalous dimension. We call this functional form "Dawson's
Sudakov". Our formalism is widely applicable for the resummation of subleading
power rapidity logarithms in other more phenomenologically relevant
observables, such as the EEC in QCD, the $p_T$ spectrum for color singlet boson
production at hadron colliders, and the resummation of power suppressed
logarithms in the Regge limit.
