We consider the unique measure of maximal entropy for proper 3-colorings of $\mathbb {Z}^{2}$ , or equivalently, the so-called zero-slope Gibbs measure. Our main result is that this measure is Bernoulli, or equivalently, that it can be expressed as the image of a translation-equivariant function of independent and identically distributed random variables placed on $\mathbb {Z}^{2}$ . Along the way, we obtain various estimates on the mixing properties of this measure.

Consider an ergodic Markov chain on a countable state space for which the return times have exponential tails. We show that the stationary version of any such chain is a finitary factor of an independent and identically distributed (i.i.d.) process. A key step is to show that any stationary renewal process whose jump distribution has exponential tails and is not supported on a proper subgroup of ${\mathbb {Z}}$ is a finitary factor of an i.i.d. process.