We will discuss a new point of view on long range dependence, that does not rely on the rate of decay of corelations. Rather, the passage between short and long memory is seen as a phase transition in parameter space. In this case we will see such a phase transition for stationary stable processes. These can be viewed as parametrized by non-singular flows on a sigma-finite measure space, plus a kernel, (and what is called a cocycle). It turns out that if the flow is dissispative, then the maximum of the process grows at the rate of $n^{1/\alpha}$, while if the flow is conservative, then it grows at a strictly slower rate. We will see limit theorems and examples.