## What do these fixed points have in common?

Posted by Eric Kidd Thu, 12 May 2011 12:09:00 GMT

A question asked while standing in the shower: What do all of the following have in common?

- Banach and Brouwer fixed points. If you’re in Manhattan, and you crumple up a map of Manhattan and place it on the ground, at least one point on your map will be exactly over the corresponding point on the ground. (This is true even if your map is
*larger*than life.) - The fixed points computed by the Y combinator, which is used to construct anonymous recursive functions in the lambda calculus.
- The Nash equilibrium, which is the stable equilibrium of a multi-player game (and one of the key ideas of economics). See also this lovely—if metaphorical—rant by Scott Aaronson.
- The eigenvectors of a matrix, which will still point in the same direction after multiplication by the matrix.

At what level of abstraction are all these important ideas really just the same idea? If we strip everything down to generalized abstract nonsense, is there a nice simple formulation that covers all of the above?

(I can’t play with this shiny toy today; I have to work.)