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.
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An arbitrary bijection could just scramble the points other than the barycentres in any what it pleased. Is the function required to be continuous, or piecewise linear or what? If it's piecewise linear then only the identity will do. If it's just continuous, then there are lots of possibilities: try looking at a $1$-simplex, so you are asking for continuous bijections $f : [0, 1] \to [0, 1]$ that fix $0$, $1/2$ and $1$ and there are lots of those. – Rob Arthan May 14 '20 at 21:08
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Good point. I wanted it to be "monotonic" with respect to each coordinate, but now that I think about it I'm not exactly sure what that means. It's easy to visualize for the 1-simplex (i.e. (x,y) → (-2x^3+3x^2,-2y^3+3y^2)), but I don't have a good sense of what it would look like for higher dimensions. – Jack May 14 '20 at 23:41