This document provides an overview of fixed point theorems. It defines fixed points as points where a function maps to itself (f(x)=x). Brouwer's fixed point theorem states that any continuous function mapping a closed ball in Euclidean space to itself must have a fixed point. Implications include that there is always a hurricane somewhere on Earth and a method for constructing fixed points by overlaying maps. The hairy ball theorem and Kakutani's fixed point theorem are also discussed.

1. Fixed Point
Theorems
a gentle
introduction
Annual Seminar Week
IITB

2. Overview
• Some History
• What are fixed points(FP)?
• The statement of Brouwer’s Fixed point theorem
o The hairy ball theorem
• Coffee cup , hurricanes , and maps
• An interesting construction of FP’s in a very
restricted case

3. What are fixed points?
• Fixed points to a functions are the points where
f(x)=x
• Fixed point theorem’s basically say that under
certain conditions , f will have a fixed point
• And variations in these conditions give rise to
various fixed points theorems.

4. The obvious fixed point
theorem
• Every function that maps to itself in one dimension
has a fixed point (a.k.a. the Intermediate-value
theorem)
x2
x1
x1 x2

5. Generalization to n-dimensions
Brouwer’s fixed point theorem
• Every continuous function from a closed ball of a
Euclidean Space to itself has a fixed point.
• Ball => Compact , Convex (not the spherical notion)
• Euclidean Spaces -> n-dimensional spaces
obvious examples are 2d spaces , 3d spaces

6. Hairy Ball theorem
• You can't comb a hairy ball flat without creating a
cowlick!

7. Some implications
• Fixed point in a coffee cup!

8. Some implications
• There is always a hurricane somewhere on the
earth!
• This follows from the hairy ball
theorem and the fact that
wind is a continuous transform .
• Brouwer’s FPT is used by
John Nash
(“A beautiful mind”)
to prove the existence of
Nash-Equilibrium

9. Some implications
• In computer graphics we sometimes need a
continuous function that generates an orthogonal
vector to a given vector.
• The hairy ball theorem implies that there is no such
function!

10. A Stronger FP theorem
Kakutani Fixed point theorem
• Constraint in Brouwer’s FP theorem is modified such
that now the function is mapping to a subset of
itself(closed ball).

11. Geometrical Construction of
fixed point, in a map overlay
• Two maps of different sizes of a country are
arranged on a table such that one of them lies on
top of the other and is completely inside it.
• FP in this setting would be a point on the table
where both the maps point to the same location.
• We use the fact that line joining FP to the vertices
makes the same angle in both the cases with a
corresponding edge of the map
• And then some pure geometry !