Variational principles for the spectral radius

The spectral radius of a functional operator with positive coefficients generated by a set of maps (a dynamical system) is shown to be a logarithmically convex functional of the logarithms of the coefficients. This yields the following variational principle: the logarithm of the spectral radius is the Legendre transform of a convex functional $T$ defined on a set of vector-valued probability measures and depending only on the original dynamical system. A combinatorial construction of the functional $T$ by means of the random walk process corresponding to the dynamical system is presented in the subexponential case. Examples of the explicit calculation of the functional $T$ and the spectral radius are presented.
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