Level-2 Large Deviation Principle for Countable Markov Shifts Without Gibbs States

We consider level-2 large deviations for the one-sided countable full shift without assuming existence of Bowen's Gibbs state. To deal with non-compact closed sets, we provide a sufficient condition in terms inducing which ensures exponential tightness sequence Borel probability measures constructed from periodic configurations. Under this establish Large Deviation Principle. apply our results to continued fraction expansion real numbers $[0,1)$ generated by R\'enyi map, and obtain Principle, as well weighted equidistribution set quadratic irrationals equilibrium states map.