at.algebraic topology - What kind of geometric operations "scale up" cohomology?

How about we change the question to this: for what kind of spaces X are there a family of continuous maps ψ^k: X->X which act on cohomology by "scaling up"? Here is a famous and bewildering example of something sort of like this:

As you can read in the notes of Sullivan's 1970 MIT course (see esp. chapter 5), if you have an algebraic variety X defined over Q(the rationals), then you get an action of G(=absolute Galois group of Q) on a space X_et=the etale homotopy type of X.

If X is a Grassmanian variety, then

* X_et has the homotopy type of the usual complex Grassmannian (up to a "profinite completion"),

* G acts on the profinite cohomology of X_et through its abelianization G_ab=Z^*(=the units of the profinite integers),

* this action of G_ab on cohomology is by scaling.

I don't know what the motivation of the original question is, so I don't know if this kind of thing has any relation for what you want. (I really posted this to advertise the Sullivan notes, which everybody should read.)