Dual description of nonextensive ensembles

q↔1/q-duality is investigated. It explains the appearance of two equilibrium density matrices ρ and σ ρq and of two Hamiltonians H and K. A Hamiltonian H with critical entropic index qc>1 is thermodynamically stable at all positive temperatures for q in the interval 2−qc1, then H is thermodynamically stable for low enough temperatures in the interval 1/qc