  We formulate a dynamical real space renormalization group approach to
describe the time evolution of a random spin-1/2 chain, or interacting
fermions, initialized in a state with fixed particle positions. Within this
approach we identify a many-body localized state of the chain as a dynamical
infinite randomness fixed point. Near this fixed point our method becomes
asymptotically exact, allowing analytic calculation of time dependent
quantities. In particular we explain the striking universal features in the
growth of the entanglement seen in recent numerical simulations: unbounded
logarithmic growth delayed by a time inversely proportional to the interaction
strength. The particle number fluctuations by contrast exhibit much slower
growth as log(log(t)) indicating blocked particle transport. Lack of true
thermalization in the long time limit is attributed to an infinite set of
approximate integrals of motion revealed in the course of the RG flow, which
become asymptotically exact conservation laws at the fixed point. Hence we
identify the many-body localized state with an emergent generalized Gibbs
ensemble.
