Code length between Markov processes (Q1806254)

In this paper it is shown that if \(X_1\) and \(X_2\) are mixing Markov processes with entropies \(h(X_1)> h(X_2)\), then there exists a finitary coding \(\varphi: X_1\to X_2\) such that the code length is in \(L^p\) for all \(p< 4/3\), so that the expected length of the code \(\varphi\) is finite. Results of this type were given by \textit{M. Keane} and \textit{M. Smorodinsky} [Isr. J. Math. 26, 352-371 (1977; Zbl 0357.94012)] (for Bernoulli processes), and more generally by \textit{M. A. Akcoglu, A. del Junco} and \textit{M. Rahe} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 47, 305-314 (1979; Zbl 0403.28017)], but without the proof that the expectation of the code length is finite. The importance of the present paper is that it provides a proof that for mixing Markov shifts the code length has finite expectation and it is in fact an \(L^p\) function for some \(p> 1\).