partition into the orbits of a dynamical system

Question

Let $T$ be a measure-preserving invertible transformation of a Lebesgue space, and let $P$ be the partition of the Lebesgue space into the orbits of $T$. 1) Is it true that $P$ is nonmeasurable (in the sense of Rokhlin) when $T$ is ergodic ? Why ? 2) If true, is it a necessary and sufficient condition for ergodicity ? 3) Is it true that the (complete) $\sigma$-field generated by $P$ is the $\sigma$-field of $T$-invariant sets ?

Answer 1

Although it appears you've already settled matters with the information in Jon's answer, I'll offer a quick summary and elaboration.

Let $(X,\mathcal{B},\mu)$ be a Lebesgue space (set + $\sigma$-algebra + probability measure) and $P$ the partition into orbits $\mathcal{O}(x) = \{T^n x \mid n\in \mathbb{Z}\}$ for an invertible measure-preserving transformation $T$.

For (3), it's exactly as you say: $\mathcal{B}$ contains each orbit $\mathcal{O}(x)$, and since a set $A$ is $T$-invariant if and only if it is a union of complete $T$-orbits, we get that the $\sigma$-algebra generated by $P$ is precisely the collection of $T$-invariant sets. The completed $\sigma$-algebra generated by $P$ is the collection of sets that are $T$-invariant mod $0$.

Consequently the answer to (1) is yes unless a single orbit carries full measure: ergodicity implies that every element of the $\sigma$-algebra generated by $P$ has measure $0$ or $1$, and consequently this $\sigma$-algebra is equivalent mod $0$ to the trivial $\sigma$-algebra. Thus for an ergodic transformation, $P$ is measurable if and only if a single partition element has full measure, which happens exactly when $\mu$ is supported on a single periodic orbit.

Finally, for (2), you observe correctly that non-measurability of $P$ follows as soon as $\mu$ has an ergodic component that is not a periodic orbit. For a concrete example, one may consider the map $T\colon [0,1]\to[0,1]$ that takes $x$ to $2x \pmod 1$, and the measure $\mu = \frac 12(\delta_0 + \lambda)$, where $\delta_0$ is the point mass on the fixed point at $0$, and $\lambda$ is Lebesgue measure. In this case $\mu$ is non-ergodic but $P$ is non-measurable. Or, if you prefer a completely non-atomic example, you can let $\nu_p$ denote the $T$-invariant measure on $[0,1]$ that comes from the $(p,1-p)$-Bernoulli measure on the full two-shift (where $p\in (0,1)$) and let $\mu$ be any convex combination of $\nu_p$ and $\nu_q$ for $p\neq q$.

Answer 2

It seems that the nice set of notes "Measure theory through dynamical eyes" by Katok and Climenhaga here (see, for example, Section 1.4) will be useful for the first question (1). It is also mentioned that if the ergodic system has more than one orbit, then ergodicity is sufficient for nonmeasurability of the partition (2). I hope this is helpful. Perhaps Vaughan Climenhaga will see this and write a nice answer.