Geodesic
Variational principles for the spectral radius
Sbornik. Mathematics, Tome 197 (2006) no. 5, pp. 633-680

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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. Bibliography: 28 titles. 
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     author = {A. B. Antonevich and K. Zajkowski},
     title = {Variational principles for the spectral radius},
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A. B. Antonevich; K. Zajkowski. Variational principles for the spectral radius. Sbornik. Mathematics, Tome 197 (2006) no. 5, pp. 633-680. http://geodesic.mathdoc.fr/item/SM_2006_197_5_a0/