We have just seen that in a system subjected to an external field and a thermostat which maintains the system's kinetic energy at a constant value there is a phase space contraction taking place. Prior to that, we showed that a fractal repeller can form in the phase-space for a Hamiltonian system with absorbing boundaries such that trajectories hitting the boundary never re-appear in the system. Some questions naturally arise as we think about these systems, such as:

1. To what kind of structure is the phase-space for a thermostatted system contracting?

2. What are the properties of such a structure? Is it a smooth hypersurface, say, or does it have a more complicated structure?

3. How do we describe fractal attractors and/or repellers and compute the properties of trajectories which are confined to them, since these objects are typically of zero Lebesgue measure in phasespace?

In this chapter, we will show that for each of these situations there is an appropriate measure that can be used to describe the resulting sets and to compute the properties of trajectories which are confined to these sets. Throughout our discussions we will suppose that the dynamics is hyperbolic. In the thermostatted case, the system contracts onto an attractor, and the attractor is characterized by an invariant measure which is smooth along unstable directions and fractal in the stable directions. Such measures are known as SRB (Sinai–Ruelle–Bowen) measures.

For a system with escape, the invariant measure on the repeller is different from that on an attractor, because escape takes place along the expanding directions.