We prove a generalization of the individual ergodic theorem of the information theory and we apply it to give a new proof of the Shereshevsky inequality connecting the metric entropy and Lyapunov exponents of dynamical systems generated by cellular automata.
"An application of the ergodic theorem of information theory to Lyapunov exponents of cellular automata." Topol. Methods Nonlinear Anal. 40 (2) 415 - 427, 2012.