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For a finite, recurrant, invertible, measure preserving dynamical system with transformation $T$, I can show that if $T$ is ergodic, then the induced transformation for any positive-measure set is ergodic. How can I prove the converse - that is, if there exists a positive-measure set for which the induced transformation is ergodic, then $T$ is ergodic?

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  • $\begingroup$ I've to do this exercise also but with the assumption that $\mu(\cup_{k\geq 1}T^{-k} A)=1$. Didn't you needed any assumption? $\endgroup$ – simon Oct 18 '13 at 9:09
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I was searching for a similar statement and I found this. I don't know if it is OK to provide an answer to such an old thread.

Let $Y$ be a subset of $X$ such that $$ \bigcup_{n\geq 1} T^{-n}Y=X$$ Call $\varphi$ the return time function of $Y$ and $T_Y$ the first return map of $Y$. There are two main ingredients in the proof: for every measurable set $A$, we have $$T_Y^{-1}(Y\cap A)=\bigcup_{n\geq 1}Y\cap \{\varphi=n\}\cap T^{-n}A.$$ This is essentially a decomposition of $Y$ into disjoint pieces which land to $A$ for the first time at $n$. The second key idea is that if you have a $T$-invariant set $A$, it induces a $T_Y$-invariant set $A\cap Y$: $$T_Y^{-1}(Y\cap A)=\bigcup_{n\geq 1}Y\cap \{\varphi=n\}\cap T^{-n}A=\bigcup_{n\geq 1}Y\cap \{\varphi=n\}\cap A=A\cap Y.$$ Since $T_Y$ is ergodic, $Y\cap A$ or $Y\cap A^c$ have zero measure. In the first case, by invariance $T^{-n}Y\cap A$ has zero measure, and by the hypothesis on $Y$ we have $$ \mu(A)=\mu\left(\bigcup_{n\geq 1}T^{-n}Y\cap A\right)\leq 0 $$ The second case is analogous. The proof works in the infinite measure as well.

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I didn't check the details but I think the following should work:

first, you have to assume that integral the first return maps is finite. Then, the suspension of the induced transformation is, up to a positive constant, the original measure preserving transformation. If you believe this, then the result you want follows form Lemma 9.24 in the book

http://books.google.ch/books/about/Ergodic_Theory.html?id=PiDET2fS7H4C&redir_esc=y

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