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under what conditions is the limit of a sequence of ergodic functions still ergodic? are there simple counter-examples to this general statement?

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A rotation $z\mapsto e^{2\pi i\alpha} z$ as a self-map of the unit circle is ergodic wrto the length measure iff $\alpha$ is irrational. So any sequence of irrational numbers converging to a rational number produces a counterexample.

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