Recently Ott, Tomforde and Willis introduced a notion of one-sided shifts over infinite alphabets and proposed a definition for sliding block codes between such shift spaces. In this work we propose a more general definition for sliding block codes between Ott-Tomforde-Willis shift spaces and then we prove Curtis-Hedlund-Lyndon type theorems for them, finding sufficient and necessary conditions under which the class of the sliding block codes coincides with the class of continuous shift-commuting maps.

Ott, Tomforde, and Willis proposed a useful compactification for one-sided shifts over infinite alphabets. Building from their idea we develop a notion of two-sided shift spaces over infinite alphabets, with an eye towards generalizing a result of Kitchens. As with the one-sided shifts over infinite alphabets our shift spaces are compact Hausdorff spaces but, in contrast to the one-sided setting, our shift map is continuous everywhere. We show that many of the classical results from symbolic dynamics are still true for our two-sided shift spaces. In particular, while for one-sided shifts the problem about whether or not any M -step shift is conjugate to an edge shift space is open, for two-sided shifts we can give a positive answer for this question.

Chargaff's second parity rule for short oligonucleotides states that the frequency of any short nucleotide sequence on a strand is approximately equal to the frequency of its reverse complement on the same strand. Recent studies have shown that, with the exception of organellar DNA, this parity rule generally holds for double-stranded DNA genomes and fails to hold for single-stranded genomes. While Chargaff's first parity rule is fully explained by the Watson-Crick pairing in the DNA double helix, a definitive explanation for the second parity rule has not yet been determined. In this work, we propose a model based on a hidden Markov process for approximating the distributional structure of primitive DNA sequences. Then, we use the model to provide another possible theoretical explanation for Chargaff's second parity rule, and to predict novel distributional aspects of bacterial DNA sequences.

Recently a generalization of shifts of finite type to the infinite alphabet case was proposed, in connection with the theory of ultragraph C*-algebras. In this work we characterize the class of continuous shift commuting maps between these spaces. In particular, we prove a Curtis-Hedlund-Lyndon type theorem and use it to completely characterize continuous, shift commuting, length preserving maps in terms of generalized sliding block codes.MSC 2010: 37B10, 54H20, 37B15

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