Abstract
A definition of hyperbolic meromorphic functions is given and thenwe discuss the dynamical behavior and the thermodynamic formalism ofhyperbolic functions on their Julia sets. We prove the importantexpanding properties for hyperbolic functions on the complex planeor with respect to the Euclidean metric. We establish the Bowenformula for hyperbolic functions on the complex plane, that is, thePoincare exponent equals to the Hausdorff dimension of the radialJulia set and furthermore, we prove that all the results in theWalters' theory hold for hyperbolic functions on the Riemann sphere.
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