Variational properties of multifractal spectra

For hyperbolic diffeomorphisms, we describe the variational properties of t he dimension spectrum of equilibrium measures on locally maximal hyperbolic sets, when the measure or the dynamical system are perturbed. We also obta in explicit expressions for the first derivative of the dimension spectra a nd the associated Legendre transforms. This allows us to establish a local version of multifractal rigidity, i.e. of a 'multifractal' classification o f dynamical systems based on their multifractal, spectra.