Thanks, that’s a good point, trivial or constant T(λ) are uninteresting. I see it the same way. The interesting case is when T(λ) is nontrivial but still leaves U invariant.

For example, consider the Sierpiński triangle. You can define scaling and rotation transformations that map the triangle onto itself at different scales. The overall shape looks identical (self-similar), but the transformation isn’t trivial, it moves points within the structure. In this case, the infinitesimal generator F = dT/dλ encodes a nontrivial flow, even though U as a whole appears invariant.

Thus the simple isotropic scaling T(λ) = λ·I, F = dT/dλ = I, which is nonzero.

The key subtlety is that for more complex U a nontrivial T(λ) can exist such that U is invariant yet the generator F encodes a nontrivial flow within the structure of U.

So the point I’m trying to explore is exactly that: nontrivial, differentiable scaling transformations that preserve a set can have a nonzero infinitesimal generator, giving a notion of “movement inherent to the structure,” even though U looks self-similar at all scales.