Variational principles for spectral radii of positive functional operators

Functional operators, i.e., sums of weighted shift operators generated by various maps, are considered. For functional operators with positive coefficients, variational principles for spectral radii are obtained. These principles say that the logarithm of the spectral radius is the Legendre transform of a certain convex functional $T$ defined on the set of probability vector-valued measures and depending on the original dynamical system and the functional space considered. In the subexponential case, we obtain the combinatorial structure of the functional $T$ with the help of the corresponding random walk process constructed according to the dynamical system.