Recurrence and ergodicity for non-measure preserving transforms

Question

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?

Answer 1

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.

Answer 2

Disclaimer: I don't know if there are corresponding notions for non measure preserving systems. These are purely my thoughts on the subject which have nothing to do with any deep mathematical concept. It's a clearly a heuristic approach but this is why I think it doesn't really make sense that they exist. I might be dead wrong though. I apologize for the long post.

The milestone of ergodic theory is to be able to compute the limiting behaviour of discrete averages of a function and we would like to compare it the "average" of the function which in a broader way is the integral of the function. This is where measure is "needed" . Now, we want this measure to be invariant under $T$ because we want the pushforward and pullback (under $T$) measure space to be the same, in order to basically measure how $T$ acts on that space. Now ergodicity is a subcategory of such a system, that basically says that you cannot measurably further break down your system into measure preserving subsystems.