Ergodic Theory Meets Polarization. I: An Ergodic Theory for Binary Operations

Understand

An open problem in polarization theory is to determine the binary operations that always lead to polarization (in the general multilevel sense) when they are used in Ar{\i}kan style constructions.

• This paper, which is presented in two parts, solves this problem by providing a necessary and sufficient condition for a binary operation to be polarizing.
• This (first) part of the paper introduces the mathematical framework that we will use in the second part to characterize the polarizing operations.
• We define uniformity preserving, irreducible, ergodic and strongly ergodic operations and we study their properties.