  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 $\alpha>1$
describes the contraction process, while the dual index $\alpha ^{\prime
}=2-\alpha<1$ 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.
