[This question][1] is a very nice topological one, a cousin of Brouwer's fixed point theorem and related to several questions in the literature:

> Suppose that $f$ and $g$ are two commuting continuous mappings from the closed unit disk to itself. Does there always exist a point $x$ such that $f(x)=g(x)$? 


  [1]: https://mathoverflow.net/questions/3332/two-commuting-mappings-in-the-disk