Entropies for severely contracted configuration space

We demonstrate that dual entropy expressions of the Tsallis type apply naturally to statistical–mechanical systems that experience an exceptional contraction of their configuration space. The entropic index α>1 describes the contraction process, while the dual index α′=2−α<1 defines the contraction dimension at which extensivity is restored. We study this circumstance along the three routes to chaos in low-dimensional nonlinear maps where the attractors at the transitions, between regular and chaotic behavior, drive phase-space contraction for ensembles of trajectories. We illustrate this circumstance for properties of systems that find descriptions in terms of nonlinear maps. These are size-rank functions, urbanization and similar processes, and settings where frequency locking takes place.