Towers, Conjugacy and Coding

We consider three theorems in ergodic theory concerning a fixed aperiodic measure preserving transformation σ of a Lebesgue probability space (X,A, μ) and show that these theorems are all equivalent. Two of these results concern the existence of a partition of the space X with special properties. The third theorem asserts that the conjugates of σ are dense in the uniform topology on the space of automorphisms. The first partition result is Alpern’s generalization of the Rokhlin Lemma, the so-called Multiple Rokhlin Tower theorem stating that the space can be partitioned into denumerably many columns and the measures of the columns can be prescribed in advance; the second partition result is a coding result which asserts that any mixing Markov chain can be represented by σ and some partition of the space indexed by the state space of the Markov chain.