Lyapunov partition functions for the dimensions of chaotic sets

Multifractal dimension spectra for the stable and unstable manifolds of invariant chaotic sets are studied for the case of invertible two-dimensional maps. A dynamical partition-function formalism giving these dimensions in terms of local Lyapunov numbers is obtained. The relationship of the Lyapunov partition functions for stable and unstable manifolds to previous work is discussed. Numerical experiments demonstrate that dimension algorithms based on the Lyapunov partition functions are often very efficient. Examples supporting the validity of the approach for hyperbolic chaotic sets and for nonhyperbolic sets below the phase transition (q<$q_T$) are presented.