  The $SU(N)$ Yang-Mills theory in $\mathbb R^4\times S^1$ spacetime is studied
as a simple toy model of Gauge-Higgs unification. The theory is perturbatively
nonrenormalizable but could be formulated as an asymptotically safe theory,
namely a nonperturbatively renormalizable theory. We study the fixed point
structure of the Yang-Mills theory in $\mathbb R^4\times S^1$ by using the
functional renormalization group in the background field approximation. We
derive the functional flow equations for the gauge coupling and the background
gauge-field potential. There exists a nontrivial fixed point for both couplings
at finite compactification radii. At the fixed point, gauge coupling and vacuum
energy are both relevant. The renormalization group flow of the gauge coupling
describes the smooth transition between the ultraviolet asymptotically safe
regime and the strong interacting infrared limit.
