  We study the phase structure of five-dimensional Yang-Mills theories coupled
to Dirac fermions. In order to tackle their non-perturbative character, we
derive the flow equations for the gauge coupling and the effective potential
for the Aharonov-Bohm phases employing the Functional Renormalisation Group. We
analyse the infrared and ultraviolet fixed-point solutions in the flow of the
gauge coupling as a function of the compactification radius of the fifth
dimension. We discuss various types of trajectories which smoothly connect both
dimensional limits. Last, we investigate the phase diagram and vacuum structure
of the gauge potential for different fermion content.
