Average redundancy of the Shannon code for Markov sources

It is known that for memoryless sources, the average and maximal redundancy of fixed-to-variable length codes, such as the Shannon and Huffman codes, exhibit two modes of behavior for long blocks. It either converges to a limit or it has an oscillatory pattern, depending on the irrationality or rationality, respectively, of certain parameters that depend on the source. Here, we extend these findings for the Shannon code to the case of a Markov source, which is considerably more involved. We provide a precise characterization of the redundancy of the Shannon code redundancy for a class of irreducible, periodic and aperiodic Markov sources.