We consider a family of Gibbs states describing the multifractal spectrum of local entropies. An accumulation point μ∞ of this sequence may be called a Gibbs ground state or a zero temperature limit, because the interpretation of q as the inverse of the temperature. We prove that, in more general systems than symbolics, that μ∞ is a maximizing measure. We also do geometric and combinatorial approaches as well as some interpretations in a particular case.