  Dynamical maps describe general transformations of the state of a physical
system, and their iteration can be interpreted as generating a discrete time
evolution. Prime examples include classical nonlinear systems undergoing
transitions to chaos. Quantum mechanical counterparts show intriguing phenomena
such as dynamical localization on the single particle level. Here we extend the
concept of dynamical maps to an open-system, many-particle context: We
experimentally explore the stroboscopic dynamics of a complex many-body spin
model by means of a universal quantum simulator using up to five ions. In
particular, we generate long-range phase coherence of spin by an iteration of
purely dissipative quantum maps. We also demonstrate the characteristics of
competition between combined coherent and dissipative non-equilibrium
evolution. This opens the door for studying many-particle non-equilibrium
physics and associated dynamical phase transitions with no immediate
counterpart in equilibrium condensed matter systems. An error detection and
reduction toolbox that facilitates the faithful quantum simulation of larger
systems is developed as a first step in this direction.
