Existence of a linear Pfaff system with arbitrary bounded disconnected lower characteristic set of positive Lebesgue \(m\)-measure (Q517140)

The authors study linear Pfaffian systems of the form \(\partial x/\partial t_i=A_i(t)x\) where \(x\) is an \(n\)-dimensional and \(t\) an \(m\)-dimensional vector which are assumed to be integrable. Lower characteristic vectors describe asymptotic properties of non-trivial solutions and the lower characteristic set of a system is defined as the union of all such vectors of all non-trivial solutions. In earlier works, the existence of such systems with a lower characteristic set of positive Lebesgue measure was mainly shown in the case \(m=2\). In this work, the authors consider the case of arbitrary \(m\geq2\). They prove that given an arbitrary bounded closed domain consisting of a finite number of disjoint components satisfying some further conditions there exists an integrable Pfaffian system with bounded smooth coefficient matrices such that its lower characteristic set coincides with the given set.