fixed point roulette
try to escape brouwer
can you beat a theorem?
Brouwer's fixed point theorem: any continuous function from a convex compact set to itself has a fixed point -- a point that maps to itself.
your mission: define continuous maps with no fixed points. the theorem says you can't.
drag the control points to shape f : [0,1] → [0,1]. make the curve avoid the dashed line y = x.
drag the colored dots to define where each corner of the unit square maps to. the warped grid shows your continuous map.
same game, different shape. rotate the annulus and check for fixed points.
the math
brouwer's fixed point theorem
the 1D proof (IVT)
for f : [0,1] → [0,1] continuous, define g(x) = f(x) − x. then g(0) = f(0) ≥ 0 and g(1) = f(1) − 1 ≤ 0. by the intermediate value theorem, g has a zero. that zero is the fixed point.
why continuity matters
define f(x) = x + 1/2 (mod 1). this maps [0,1] to itself, and no point is fixed: every point shifts by 1/2. but f is discontinuous at x = 1/2. without continuity, the curve can "jump" over the diagonal y = x.
why the domain matters
a rotation of the annulus maps it to itself continuously with no fixed point. the center of rotation would be fixed, but it is not in the domain. the annulus has a hole, so it is not convex (in fact, not even contractible). Brouwer needs the domain to be at least contractible.
applications
- nash equilibrium. every finite game has a mixed-strategy equilibrium. Nash's 1950 proof uses Brouwer (via Kakutani's generalization to set-valued maps).
- general equilibrium. Arrow and Debreu proved competitive equilibrium exists in any economy satisfying mild conditions. the core step is a fixed-point argument.
- differential equations. the Schauder fixed point theorem extends Brouwer to infinite dimensions, proving existence of solutions to many PDEs.
- the crumpled map theorem. crumple a map of a city and place it on a flat copy. at least one point on the crumpled map sits directly above its real location on the flat one.