We present a series of numerical and analytical computations on heat conduction for a strongly chaotic system—the Lorentz gas. Heat conduction is characterized by nontrivial features: While the heat conductivity is well defined in the thermodynamic limit, a linear gradient appears only for quite small temperature differences. The key dynamical feature inducing such a behavior is recognized as deterministic diffusion (along transport direction) which is usually associated to full hyperbolicity.

We consider the motion of a point particle in right triangular billiards. By considering the global dynamics (when acute angles are not rationally connected to π), or the discrete reduced dynamics (when acute angles are rational multiples of π), we find numerical evidence for the conjecture that the motion is ergodic and weakly mixing. These dynamical features are intimately related to nontrivial scaling properties of the spectrum of the evolution operator.

We consider a kicked system on the cylinder obtained upon quantization of a chaotic area-preserving map. We use the thermodynamic formalism to investigate the scaling properties of the fractal spectrum. In time evolution we observe anomalous diffusion with an exponent closely related to the Hausdorff dimension of the spectrum, and dependent upon the parameters of the system.

The kicked Harper model is numerically investigated. The long-time nature of the wave function dynamics and a multifractal analysis of the spectrum provide evidence for the existence of different regimes marked by different spectral types and diffusion exponents, with a phase diagram substantially different from that of the standard Harper model.