The book Cycle Representations of Markov Processes solves the problem of Mapping Stochastic Matrices induced from a Markov Chain into Partitions using a $\lambda$-preserving ($\lambda$ is a Lebesgue Measure) transformation of the interval $f_{t} = (x+t) \bmod t$ ( $t$ is the rotational length to be considered is $\frac{1}{n!}$) but don't explain if the partition is a Markov Partition or a Generating Partition. I searched several references looking for this proof but there is nothing written about these partitions. If these particular partitions are Markov, how to prove this fact?
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