I would like to know if there are sufficient criteria for the composition of two ergodic maps to be still ergodic.

My context is piecewise affine transformations of the torus in arbitrary dimensions

  • $\begingroup$ In general, I don't think there are useful sufficient conditions. $\endgroup$ – Anthony Quas Oct 21 '15 at 14:55
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    $\begingroup$ For commuting circle shifts, your question boils down to asking for sufficient criteria for the sum of two irrational numbers to be irrational. It seems difficult to come up with any usable criteria other than the tautological "the sum is not rational". $\endgroup$ – Terry Tao Oct 21 '15 at 16:08
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    $\begingroup$ @TerryTao this is actually a nice condition IMO. One can well imagine that extending the question to a larger family of maps including non-commuting maps, maps on higher dimensional tori etc, might lead to some very nice number-theoretic conditions analogous to your "the sum is irrational" condition. I agree however that at the level of generality asked for by the OP there is unlikely to be a useful, non-tautologically obvious condition. $\endgroup$ – Dan Romik Oct 21 '15 at 19:01
  • $\begingroup$ Sorry for pinging you on an unrelated question - I just wanted to let you know that there is a discussion on meta related to one of your questions. (That question is now deleted, so I was not able to ping you there.) $\endgroup$ – Martin Sleziak Sep 19 '17 at 4:04

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