In this paper we discuss a simple deterministic model for a field driven, thermostated random walk that is constructed by a suitable generalization of a multibaker map. The map is a usual multibaker, but perturbed by a thermostated external field that has many of the properties of the fields used in systems with Gaussian thermostats. For small values of the driving field, the map is hyperbolic and has a unique Sinai-Ruelle-Bowen measure that we determine analytically to first order in the field parameter. We then compute the positive and negative Lyapunov exponents to second order and discuss their relation to the transport properties. For higher values of the parameter, this system becomes nonhyperbolic and possesses an attractive fixed point. @S1063-651X~99!03801-5# In the past several years a great deal of attention has been devoted to computer and analytic studies of the chaotic properties of fluid systems subjected to external fields and to Gaussian thermostats which maintain a constant kinetic or total energy in the system, in the presence of the field @1,2#. The interest in this subject stems not only from the method’s value as a means of simulating nonequilibrium flows and computing their properties, but also because there is a connection between transport properties, nonequilibrium fluctuations, and the underlying microscopically chaotic properties of the fluid. This connection has been explored from computational @3‐8# and analytic @9‐13# points of view. The purpose of this paper is to describe a model system in which the transport and dynamics of a thermostated system can be studied in great detail, and in which one can explicitly construct the Sinai-Ruelle-Bowen ~SRB! measure @14# and describe such properties as the transition from hyperbolic to nonhyperbolic behavior, and related phenomena. These properties have been explored in previous work @10,15#, but have not yet been studied in great detail, due either to the complexity or to the simplicity of the models treated up till now @11‐13#. The model discussed here allows one to gain some insights into the general class of properties of thermostated systems, while keeping the analytical and computational difficulties to manageable proportions. It is one of the few cases known so far where one can check some of the general properties of thermostated systems on a specific model. The model we consider is a variant of the multibaker maps studied by Tasaki and Gaspard @16‐18#, which are deterministic models for the diffusion of a particle on a onedimensional lattice. The map considered here has, in addition, an external driving field which is constructed so as to model the effect of a thermostated electric field on charged particles in a two-dimensional setting. We present the model and then calculate the chaotic properties at small values of the external field. We obtain an expression for the stationary state SRB measure to first order in the applied field, and the positive and negative Lyapunov exponents to second order in the applied field. This allows us to verify the interesting re