MultiTowers, conjugacies and codes: Three theorems in ergodic theory, one variation on Rokhlin's Lemma

We show that three theorems about the measurable dynamics of a fixed aperiodic measure preserving transformation $\tau$ of a Lebesgue probability space $(X, \mathcal{A}, \mu)$ are equivalent. One theorem asserts that the conjugates of $\tau$ are dense in the uniform topology on the space of automorphisms. The other two results assert the existence of a partition of the space $X$ with special properties. One partition result shows that given a mixing Markov chain, there is a code (i.e., a partition of the space) so that the itinerary process given by $\tau$ and the partition has the distribution of the given Markov Chain. The other partition result is a generalization of the Rokhlin Lemma, stating that the space can be partitioned into denumerably many columns and the measures of the columns can be prescribed in advance. Thus the first two results are equivalent to this strengthening of Rokhlin's Lemma.