Is there a definition of recurrence and ergodicity for a transformation $T$ that is not measure preserving? All the definitions of recurrence and ergodicity I have found have always been in concern of measure preserving transformations. Why is this?
Recurrence and ergodicity for non-measure preserving transforms
dynamical systemsergodic-theory
Best Answer
Perhaps the main reason for this requirement is the connection with the ergodic theorem, which has as a hypothesis that $T$ is measure preserving. Using that theorem, it follows that $T$ is ergodic if and only if for each real valued $L^1$ function $f$ its time average $$\hat f(x) = \lim_{n \to \infty} \frac{1}{n} \sum_{k=0}^{n-1} f(T^k x) $$ is constant almost everywhere (the statement of the ergodic theorem is that $\hat f(x)$ exists almost everywhere, and defines a $T$-invariant $L^1$ function).
Nonetheless, ergodicity is studied in a more general setting using the concept of a quasi-invariant measure, meaning a measure $\mu$ such that $T_*(\mu)$ and $\mu$ have the same measure zero subsets. The definition of ergodicity then applies in this situation, and there are many interesting examples and applications. One of my favorite examples is that the fractional linear action of $SL_2(\mathbb Z)$ on $\mathbb R \cup \{\infty\}$ is ergodic with respect to Lebesgue measure; this is not a single transformation, in fact it's a whole group action, but the concepts apply nonetheless.