Invariants for detecting changes in a system of linear processes

摘要

In the present article, we investigate a possible invariant to describe changes of volumes by a system of invertible linear operators. In the single operator case, the volume change is measured by the logarithm of the absolute value of the determinant and we try to generalize this invariant to multiple operator cases. We start with naive but intuitive assumptions for an invariant to satisfy and we obtain a universal invariant by taking a tensor power of real numbers as its target group. An example of a continuous invariant to the real numbers is given. We also formulate more appropriate mathematical assumptions for an invariant of volume change for the systems of commuting invertible linear operators to satisfy and obtain a universal invariant by introducing a factor group of a Milnor's K-group of real numbers as its target group. But, in this formulation, there is no nontrivial continuous invariant to the real numbers and the invariants are understood as abstract mathematical maps into direct sum of copies of Q, the rational numbers.