  In the present paper are analysed the conditions for the validity of the
Tsallis Statistics. The same have been done following the analogy with the
traditional case: starting from the microcanonical description of the systems
and analysing the scaling properties of the fundamental macroscopic observables
in the Thermodynamic Limit. It is shown that the Generalized Legendre Formalism
in the Tsallis Statistic only could be applied for one special class of the
bordering systems, those with non exponential growth of the accessible states
density in the thermodynamic limit and zero-order divergency behavior for the
fundamental macroscopic observables, systems located in the chaos threshold.
