Ergodic Theory Meets Polarization I: A Foundation of Polarization Theory

An open problem in polarization theory is to determine the binary operations that always lead to polarization when they are used in Arıkan style constructions. This paper solves this problem by providing a necessary and sufficient condition for a binary operation to be polarizing. The characterization is given in terms of a new mathematical framework that we introduce. We show that a binary operation is polarizing if and only if its inverse is strongly ergodic.