Let $K$ be a finite simplicial complex and $f:K\to K$ a periodic (finite order) homeomorphism. Using the simplicial approximation theorem, is it possible to obtain a simplicial approximation $g:K\to K$ such that $f$ has a fixed point $x\in K$ if and only if $g$ has a fixed point for the same $x$?
It might be a stretch, but I've been told that a simplicial approximation can approximate $f$ arbitrarely close (as long as we subdivide $K$ enough). I can't find any source for this however, so is this even true? And in this case, does this mean we can take the approximation 'close enough' so it imitates the fixed points?
Thanks in advance for the help!