Where can I find a detailed proof of the following proposition?

Question

Consider the following proposition:

Proposition: If $p$ is a periodic point of period $m$ for a $C^1$ map $f$ and the differential $Df_p^m$ does not have 1 as an eigenvalue then for every map $g$ sufficiently close to $f$ in the $C^1$ topology there exists a unique periodic point of period $m$ close to p.

Can someone tell me where can I find a detailed proof of this Proposition? Thank you!

Answer 1

It is a simple consequence of the implicit function theorem on Banach spaces:

Consider the map $F(g,q)=g^m(q)-q$ and compute the Fréchet derivative $\partial_q F(f,p)$.