Correlational properties of the full-length sequences based on the discretized Markov transformations

We have previously defined the discretized Markov transformations and the full-length sequences based on such transformations. In view of basic properties of the normalized cross- and auto-correlation functions for the de Bruijn sequences that can be regarded as the full-length sequences based on the discretized dyadic transformation, we obtain correlational properties of the full-length sequences based on the discretized golden mean transformation. We generalize this result and give the correlational properties of the discretized Markov β-transformations with the alphabet Σ = {0,1, ···, k - 1} and the set F = {(k - 1)(k - 1)} of forbidden blocks (k ≥ 2), whose underlying transformations exhibit the most fundamental class of greedy β-expansions of real numbers.