I would need to reference the following seemingly very well known fact:



> 
> If f:$M\to M$ is a diffeomorphism of finite order, then at any point in the fixed-point set of f the manifold M has coordinates with respect to which f is linear.
> 
> 
> 


I've seen it called "local linearization theorem" in some lecture notes, but without reference to a textbook.


Can somebody suggest a standard reference for that?



Reference for local linearization theorem
S. Bochner, *Compact groups of differentiable transformations*, **Ann. of Math.** (2) 46 (1945), 372–381. MR MR0013161 (7,114g)


