## Brouwer fixed-point theorem and invariant measure density

1.How to prove Brouwer fixed-point theorem in two dimensions? Assuming f is one-to-one mapping, S is a square and f(S) is contained in S, then there is a fixed point in S.

2. Consider the map
$f(x) = \begin{cases}ax+c, 0\leq x \leq c\\a-ax, c\leq x \leq 1 \end{cases}$

where $a =\frac{(\sqrt {5}+1) }{2}$ and $c=\frac{1}{(a+1)}$

We know that an invariant measure $\mu$ of the map f has a density p(x) which is constant on [0,c] and also on [c,1]. Find p(x)

My idea is $\mu(S)=$length of S, where S is contained in the unit interval[0,1].The density is simply p(x)=1 for x in that interval. But i dont know how to use the conjugacy to find the density of the invariant measure.
Thank you.