The Lyapunov spectrum as the Newton-Raphson method for countable Markov interval maps

We consider MRL maps (Markov-Renyi-Lüroth), a class of interval maps with infinitely many branches that can have parabolic fixed points. We prove that for every MRL map $T$, the Lyapunov spectrum can be expressed in terms of the Legendre transform of the topological pressure of $-t\log \mathrm{\mid }{T}^{\mathrm{\prime }}\mathrm{\mid }$, generalizing previous results in the area. We also show that the Lyapunov spectrum coincides with a function directly related to the Newton-Raphson method applied to the topological pressure of $-t\log \mathrm{\mid }{T}^{\mathrm{\prime }}\mathrm{\mid }$.