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.