Poincaré-Bendixson's biquadratic local conditioning with respect to geometric probability theory

Abstract

We are to formulate the invariant measures to fix compact subsets on the fluctuated situations for two autonmous systems in transition. Such a kind of two autonomous systems is formulated and intermediated by a certain formalism of their relevant non autonomous system that cannot be revealed other than by etale cohomology, l-adic cohomology and L-funtions as the functionality of transmutations amidst them, that is, certain elements fixed in their integrability and in recurrence for their definition is called probabilities in geometric probability theory by its local biquadratic conditioning after Poincaré-Bendixson.