  In the case of a system with an unbounded hamiltonian the entropic index q of
non-extensive thermodynamics has an upperbound q_c>1 beyond which the formalism
becomes meaningless. The expression 1/(q_c-1) is the dimension of the state
space (i.e. the manifold of density matrices) in the context of non-commutative
geometry. For q=q_c an ultraviolet cutoff E< E_ub is needed to guarantee the
existence of the equilibrium density matrix. Duality between q>1 and
q<1-statistics via a q --> 1/q transformation is established. It leads to an
overall picture in which the meaning of both q>1-statistics and q<1-statistics
is clarified.
