Large deviation principles for periodic points of the Dyck shift

We investigate periodic points of the Dyck shift from the viewpoint of large deviations. We establish the level-2 Large Deviation Principle with the rate function given in terms of Kolmogorov-Sinai entropies of shift-invariant Borel probability measures. Unlike topologically mixing Markov shifts, the level-2 rate function is non-convex and level-1 rate functions are superpositions of two convex continuous functions. Using the thermodynamic formalism, we show the analyticity of level-1 rate functions in some relevant cases. We display a non-convex level-1 rate function with a non-differentiable point in the interior of its effective domain.