An example of a factor map without a saturated compensation function

Abstract

Let (X, S), (Y, T) be topological dynamical systems and rr : X --> Y a factor map. A function F is an element of C(X) is a compensation function if P(F + phi circle pi) = P(phi) for all phi is an element of C(Y). We present an example of a factor map pi : X --> Y between subshifts of finite type X, Y that does not have a saturated compensation function and an example of a non-Markovian factor map with a saturated compensation function. Also we provide a necessary and sufficient condition for a certain type of factor map to have a saturated compensation function.