The strange sets which arise in deterministic low-dimensional dynamical systems are analysed in terms of (unstable) cycles and their eigenvalues. The general formalism of cycle expansions is introduced and its convergence discussed.

Abstract. Cycle expansions are applied to a series of low-dimensional dynamically generated strange sets: the skew Ulam map, the period-doubling repeller, the Hbnon-type strange sets and the irrational winding set for circle maps. These illustrate various aspects of the cycle expansion technique; convergence of the curvature expansions, approximations of generic strange sets by self-similar Cantor sets, effects of admixture of non-hyperbolicity, and infinite resummations required in presence of orbits of marginal stability. A new exact and highly convergent series for the Feigenbaum 6 is obtained.

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. [S0031-9007(99)08614-7] PACS numbers: 44.10. + i, 05.20. -y, 05.45. -a

We claim that looking at probability distributions of finite time largest Lyapunov exponents, and more precisely studying their large deviation properties, yields an extremely powerful technique to get quantitative estimates of polynomial decay rates of time correlations and Poincaré recurrences in the -quite delicate-case of dynamical systems with weak chaotic properties.

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We introduce a cycle-expansion (fully deterministic) technique to compute the asymptotic behavior of arbitrary order transport moments. The theory is applied to different kinds of one-dimensional intermittent maps, and Lorentz gas with infinite horizon, confirming the typical appearance of phase transitions in the transport spectrum.

The Gillis model, introduced more than 60 years ago, is a non-homogeneous random walk with a position-dependent drift. Though parsimoniously cited both in physical and mathematical literature, it provides one of the very few examples of a stochastic system allowing for a number of exact results, although lacking translational invariance. We present old and novel results for this model, which moreover we show represents a discrete version of a diffusive particle in the presence of a logarithmic potential.