Equilibrium measures and Cramer asymptotics in a non-invertible dynamical system with power-law mixing

We consider a dynamical system generated by a shift in the space of finite-valued one-sided sequences. We study spectral properties of Perron–Frobenius operators associated with this system, whose potentials on the number of the term of the sequence have power-law dependence. Using these operators, we construct a family of equilibrium probability measures in the phase space having the property of power-law mixing. For these measures we prove a central limit theorem for functions in phase space and a Cramer-type theorem for the probabilities of large deviations.
Similar results for the significantly simpler case of exponential decay in the dependence of the potentials on the number of the term of the sequence were previously obtained by the author.