# Factors in probability and ergodic theory

Sinai’s factor theorem states that an ergodic stationary system of positive entropy has any independent and identically distributed (i.i.d.) system of no greater entropy as a factor. In particular, as a deterministic function of a bi-infinite sequence of i.i.d. three-sided fair dice roll, we can produce a bi-infinite sequence of i.i.d. fair coin flips; furthermore, the deterministic function is equivariant, so that if the sequence of dice roll is shifted, then its output under the function is given by shifting the original output. We will discuss factor maps in probability and ergodic theory, and recent work with Zemer Kosloff, where we prove a version of Sinai’s theorem where the input is given by independent, but not identical dice roll.