  We consider periodic Markov chains with absorption. Applying to iterates of
this periodic Markov chain criteria for the exponential convergence of
conditional distributions of aperiodic absorbed Markov chains, we obtain
exponential estimates for the periodic asymptotic behavior of the semigroup of
the Markov chain. This implies in particular the exponential convergence in
total variation of the conditional distribution of the Markov chain given
non-absorption to a periodic sequence of limit measures and we characterize the
cases where this sequence is constant, which corresponds to the cases where the
conditional distributions converge to a quasi-stationary distribution. We also
characterize the first two eignevalues of the semigroup and give a bound for
the spectral gap between these eigenvalues and the next ones. Finally, we give
ergodicity estimates in total variation for the Markov chain conditioned to
never be absorbed, often called Q-process, and quasi-ergodicity estimates for
the original Markov chain.
