Ergodicity in Transformation Implied by Ergodicity in Induced Transformation

Question

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?

Answer 1

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