Multifractal analysis in symbolic dynamics and distribution of pointwise dimension for g-measures

We introduce a multifractal formalism for potentials defined on shift syste ms. We prove that the multifractal spectra are a Legendre transform of ther modynamic functions involving the potentials studied. We obtain the fractal distribution of pointwise dimension for g-measures. Such measures are equi librium states of potentials not necessarily Holder continuous and generali ze Gibbs measures. In connection with phase transition, we also give exampl es of potentials with a non-unique equilibrium state and non-analytic multi fractal spectra.