Fixed Point Theorems
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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.
Fixed Point Theorems
- 1. Fixed Point Theorems a gentle introduction Annual Seminar Week IITB
- 2. Overview Some History What are fixed points(FP)? The statement of Brouwers Fixed point theorem o The hairy ball theorem Coffee cup , hurricanes , and maps An interesting construction of FPs in a very restricted case
- 3. What are fixed points? Fixed points to a functions are the points where f(x)=x Fixed point theorems 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 Brouwers 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 . Brouwers 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 Brouwers 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 !