Looking for a bijective function mapping an n-simplex to itself

Question

As part of a research question I am exploring, I need to find a bijective function on an n-simplex that maps the midpoint of each sub-simplex to itself. This includes all vertices, midpoints of edges, midpoints of faces, etc. The function should also be bijective on each sub-simplex. The identity map is a trivial solution, but I'd like to find a more general one. Any guidance would be appreciated, even a solution for a 2-simplex or a way to visualize potential solutions using software.