  In this work we study the entanglement entropy of a uniform quantum Ising
chain in transverse field undergoing a periodic driving of period $\tau$ . By
means of Floquet theory we show that, for any subchain, the entanglement
entropy tends asymptotically to a value $\tau$ - periodic in time. We provide a
semi-analytical formula for the leading term of this asymptotic regime: It is
constant in time and obeys a volume law. The entropy in the asymptotic regime
is always smaller than the thermal one: because of integrability the system
locally relaxes to a Generalized Gibbs Ensemble (GGE) density matrix. The
leading term of the asymptotic entanglement entropy is completely determined by
this GGE density matrix. Remarkably, the asymptotic entropy shows marked
features in correspondence to some non-equilibrium quantum phase transitions
undergone by a Floquet state analog of the ground state.
