# adic periodic approximation of a dynamical system

Let $P$ be the Pascal adic transformation. The cutting and stacking construction of $P$ corresponds to a Pascal periodic approximation'' of $P$: a sequence $(P_n)$ of periodic automorphisms strongly converging to $P$ and for each $n$ the period of a point under $P_n$ is a binomial coefficient ${n \choose k}$. Does any ergodic automorphism admit such a "Pascal periodic approximation" ?

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