Legendre–Fenchel transform of the spectral exponent of polynomials of weighted composition operators

For the spectral radius of weighted composition operators with positive weight e φ T α , , acting in the spaces L p ( X , μ ) the following variational principle holds where X is a Hausdorff compact space, is a continuous mapping and τ α some convex and lower semicontinuous functional defined on the set of all Borel probability and α -invariant measures on X . In other words is the Legendre– Fenchel conjugate of ln r ( e φ T α ). In this paper we consider the polynomials with positive coefficients of weighted composition operator of the form , . We derive two formulas on the Legendre–Fenchel transform of the spectral exponent ln r ( A φ , c ) considering it firstly depending on the function φ and the variable c and secondly depending only on the function φ , by fixing c .