RECENT RESULTS FOR THE THERMOSTATTED LORENTZ GAS

We review the behaviour of a driven, thermostatted Lorentz gas. The tw o variable Poincare section confirms that the ''attractor'' fills the whole phase space, and that the associated stationary measure is ergod ic and multifractal for fields below 2.2. Such a property of the ''att ractor'' ends with either a crisis or the emergence of elliptical regi ons, followed eventually by stable orbits. We present, accurate period ic orbit expansion calculations for the diffusion coefficient, pressur e and Lyapunov exponent. The periodic orbit approach suggests the defi nition of a partition function and gives a simple explanation for a po sitive conductivity and diffusion coefficient.