Diffeomorphism-invariant spaces of smooth functions

Solution 1:

EDIT: I left the original answer below, but I think the sequence space $\mathcal{E}[bv]$ as defined in the question and its variation defined below is just the space of functions which are constant in some neighborhood of $\infty$, and I assume that this is one of the "standard" spaces Gromov mentions. Just to clarify, I used $m$ as an index since $n$ is used for the dimension of the space.

If a function $f$ is not constant in some neighborhood of $\infty$, there exists a sequence of points $y_k \to \infty$ such that $f(y_k) \ne f(y_l)$ for $k \ne l$. Now pick integers $N_k > 1/|f(y_k) - f(y_{k+1})|$, and define the sequence $(x_m)$ by concatenating $N_1$ pairs $y_1, y_2$, then $N_2$ pairs $y_2, y_3$, in general $N_k$ pairs $y_k, y_{k+1}$. By the choice of integers, the variation in the sequence $f(x_m)$ over the $N_k$ pairs $f(y_k), f(y_{k+1})$ is greater than 1 for every $k$, so the total variation is infinite.

In the case $n=1$ one could modify the definition and restrict it to monotone sequences $(x_m)$, in which case the spaces of finite $p$-variation are all distinct, which can be shown using functions of the form $f(x) = \frac{\sin x}{x^\alpha}$ for $x \ge 1$. However, I see no similar modification which would work in dimension greater than 1.

ORIGINAL ANSWER: I don't know the answer to the second question, but for the first part it seems that replacing finite variation by finite $p$-variation for $p>1$ leads to an uncountable (nested) family of invariant subspaces. I.e., instead of bv consider the sequence space $$ \mathrm{bv}^p = \left\{ (a_n) : \sum_n |a_{n+1}-a_n|^p < \infty \right\} $$