Variational Properties of Multifractal Spectra

For hyperbolic diffeomorphisms, we describe the variational properties of the dimension spectrum of equilibrium measures on locally maximal hyperbolic sets, when the measure or the dynamical system are perturbed. We also obtain explicit expressions for the first derivative of the dimension spectra and the associated Legendre transforms. This allows us to establish a local version of multifractal rigidity, i.e., of a “multifractal” classification of dynamical systems based on their multifractal spectra.