In 1996, Glasner and King compared the setting for ergodic measure preserving transformations on $[0, 1]$ ($MPT([0, 1])$) with the collection of shift invariant measures $\mu$ on $[0,1]^{\mathbb Z}$ ($SIM([0, 1]^{\mathbb Z})$) and showed that for all $S\subset MPT([0, 1])$ $S$ is a dense $\mathcal G_\delta$-set if and only if $\{\mu\in SIM[0, 1]^{\mathbb Z}: ([0,1]^{\mathbb Z}, {\mathcal C}, \mu, sh)\cong ([0, 1], {\mathcal B}, \lambda, T)\mbox{ for some }T\in S\}$ is a dense $\mathcal G_\delta$-set. D. Rudolph extended these results and formulated:

Rudolph's Thesis: All reasonable setting for the ergodic measure preserving transformations have the same generic properties.

Work of Foreman and Weiss, much of it dating to the mid-2000's, gives a general definition of a Model for the measure preserving transformations, and shows that any two models for the measure preserving transformations have the same generic sets. (The proof uses the Vaught Transform a logic technique from the 1960's.) The main results are that:

• both $MPT([0, 1])$ and $[0, 1]^{\mathbb Z}$ are models. (This reproves the Glasner-King Theorem)
• If $X$ is ANY Polish space then $SIM(X^{\mathbb Z})$ is a model for the measure preserving transformations. In particular for countable discrete sets $\Sigma$, $SIM(\Sigma^{\mathbb Z})$ is a model for the measure preserving transformations.
• Even ${\mathbb Q}SIM$s, the space of shift invariant measures on $\Sigma^{\mathbb Z}$ that give rational values to every basic open set in $\Sigma^{\mathbb Z}$, is a model.

Question: Are there any "natural" settings that have different generic properties? Finite interval exchanges are an example, but the collection of general interval exchanges (with the natural topology) is a model.

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