Deterministic chaos for Markov chains

The paper states as its main claim that for a finite-state Markov chain with all transition probabilities strictly positive, there exists an unpredictable realization whose closure (in the topology of convergence on bounded intervals) is the set of all infinite realizations, so that every finite simulation is an arc of that same realization . The setup assumes m ≥ 2 and pij > 0 for all i, j , and Theorem 3 asserts that each finite realization is an arc of a fixed unpredictable realization s* . However, the manuscript derives this arc property from the existence of an unpredictable trajectory of the similarity map φ without actually establishing that the particular unpredictable trajectory has a dense shift-orbit; unpredictability alone does not imply density. Moreover, the paper ambiguously speaks of the “closure of the unpredictable realization” rather than the closure of its shift-orbit, which is the object that can be dense in the product topology . These are material gaps in the logical chain. In contrast, the model’s solution provides a concrete construction (by concatenating all finite words) that directly proves (a) the arc property, (b) orbit-closure density in the stated topology, and (c) unpredictability, thereby fully resolving the problem as stated.