We analyze when a multifractal spectrum can be used to recover the potential. This phenomenon is known as multifractal rigidity. We prove that for a certain class of potentials the multifractal spectrum of local entropies uniquely determines their equilibrium states. This leads to a classification which identifies two systems up to a change of variables.Facultad de Ciencias ExactasInstituto de Física de Líquidos y Sistemas BiológicosFacultad de IngenieríaGrupo de Aplicaciones Matemáticas y Estadísticas de la Facultad de Ingeniería