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I am trying to understand the following paragraph from The Classification of Non-Singular Actions, Revisited, page 5 paragraph 2.

Remember that $S \in [T]$ so that for every $x\in X, S(x) = T^{n(x)}$. If $M_1$ is large compared with $n(x)$ for all but $\epsilon_1$ of the space, and compared with $M$ as well (and $\epsilon_1$ is taken small enough), the orbit segment $\{T^jx\}_{j=0}^M$ is in fact contained in some orbit segment of $\mathcal{L}_0$ for most $x$.

So, choosing $M_1$ and $\epsilon_1$ apply Rohlin's lemma to the induced transformation on $C(0)$. This gives us that $\{{T \vert_{C(0)}}^jx\}_{j=0}^{M_1}$ is an orbit segment for all but $\epsilon_1$ of $C(0)$.

But how does this imply that we have orbit segments outside of $C(0)$? I do not understand the significance of choosing $M_1$ to be large compared with $n(x)$ or $M$.

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I believe that I have the answer. I would like to hear what people think.

The objective here is to prove $\{x,Tx,\ldots,T^{M-1}x\}$ lie in the same orbit segment $\{ S^j(y)\}_{j=0}^\infty$, for all but $\epsilon$ of the space.

Given a ladder $\mathcal{L} = \mathcal{L}(\{C(i)\},N,S)$. For any $x \in X$ there exists $y_0 \in C(0)$ such that $x = S^{j_0}(y_0) = T^{n_0}(y_0)$ with $j_0 < N$.

Similarly, for $T^kx \in X$, $k \in \{1,\ldots,M\}$ there exists some $y_k \in C(0)$ such that $T^kx = S^{j_k}(y_k) = T^{n_k}(y_k)$ with $j_k < N$. So

$$ T^{n_k}(y_k) = T^k(x) = T^{n_0 + k}(y_0) $$

$$y_k = T^{n_0 + k - n_k}(y_0)$$

That is to say, each of the $y_k \in C(0)$ can be placed in the $T$-orbit of $y_0$. Rearrangement of the $y_k$ may be necessary to make the exponent of $T$ positive. Since $y_k \in C(0)$, we can further say that each of the $y_k$ can be placed in the same $T\vert_{C(0)}$ orbit of $y_0$.

Graphical representation of how the orbit $T^i(x)$ is related to the orbit of the ladder on $C(0)$

Let $M_1$ be so large that for all but $\epsilon/2M$ of the space the return time to $C(0)$ is less than $M_1$. Then for all but $\epsilon/2$ of the space, $n_k < M_1$ and $n_0 + k - n_k < M_1 + k < M + M_1$

By Rohlin's lemma we can create $\mathcal{L}_0$ which (except on a set of measure $\epsilon/2$) contains any orbit segments of length $M + M_1$. This new ladder contains $\{ y_{k}\}_{k=0}^M$ as an orbit segment for all but $\epsilon/2$ of the space $C(0)$.

Refining $\mathcal{L}$ by $\mathcal{L}_0$ gives a new ladder $\mathcal{L}_1$. By construction, the orbit of $\{S_1^j(y_0)\}_{j=0}^\infty$ contain $y_k$ because $S_1^N = T \vert_{C(0)}$, and for any $k < M$

$$ T^k(x) = S^{j_k}(y_k) \in \{S^j(y_k)\}_{j=1}^N \subset \{S_1^j(y_0)\}_{j=0}^\infty $$

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