# Properties of a function from its pullback

Edit: I have now removed the duplication previously referred to. Thank you.

Let $M$ and $N$ be smooth manifolds and $T: M \to N$ be a smooth map. Let $\mathcal{F}(M,\mathbb{R})$ (resp.$\mathcal{F}(N,\mathbb{R})$) denote the space of smooth functions from $M$ (resp. $N$) to $\mathbb{R}$ and let $F_T$ denote the pullback of $T$, i.e. the map $F_T: \mathcal{F}(N,\mathbb{R}) \to \mathcal{F}(M,\mathbb{R})$ defined by $F_T(f)= f \circ T$.

Is it possible to recover any properties or invariants of $T$ from $F_T$, such as the Dirichlet energy, winding number, etc.?

Thank you very much.

• possible duplicate of Inverse Problem for Pullback – Stefan Waldmann Nov 10 '14 at 8:46
• There are whole books devoted to studying dynamics of a topological or smooth semigroup (by that I mean studying the semigroup $\{T^k\,|\, k\geq 0, T: M \to M\}$) via it's representation on (mostly $L^2(M)$) function spaces via pullbacks. – Vít Tuček Nov 10 '14 at 17:24
• @VítTuček Would you know of any such book titles I could look up, or any keywords that could point me in this direction? Thank you. – compmath Nov 10 '14 at 19:15
• @davidbar There are many flavours (e.g. topological dynamics, ergodic theory) a and I'm no expert. I just participated in an internet seminar on ergodic theory a few years ago. The notes are here: fa.uni-tuebingen.de/lehre/isem/12th-2008-09/… I guess the question is what kind of phenomenon do you want to study (e.g. smooth, topological, stochastic, ...) and that should narrow your search down. It may very well be that this point of view through dynamical systems is not very fruitful for your purposes. – Vít Tuček Nov 10 '14 at 20:02