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Majoring in Mathematics (view source)
Revision as of 01:58, October 16, 2009
, 01:58, October 16, 2009extremely rough treatment of fixed points: brouwer's theorem, sharkovsky's theorem. please comment on the exposition, especially if you haven't seen these before!
=== Analysis ===
=== Analysis ===
# Riemann mapping theorem. State this in terms of conformal mappings of the unit disk and draw lots of pictures!
# Riemann mapping theorem. State this in terms of conformal mappings of the unit disk and draw lots of pictures!
==== Fixed points ====
Often in mathematics or physics, it is interesting to think about what happens when we apply the same function repeatedly. For example, one might take a calculator, punch in a random number, and then hit the button "[[cosine|cos]]" repeatedly. What happens when we do this? If you try it, you might see something that looks like this:1.3, 0.267499, 0.964435, 0.569881, 0.841965, 0.665998, 0.7863, 0.706469, 0.760659, 0.724382, 0.748909, ...
It looks like these are getting closer and closer to some fixed number, and if you hit the button a few more times, you'd discover that indeed these terms approach some constant around C=0.739085. This is the value of C for which cos(C) = C. There's no way to solve for this C algebraically, but we can see from the [[intermediate value theorem]] that there must be some value of C for which this is the case, since cos(0) > 0 but cos(1) < 1.
Generally, a value x is called a '''fixed point''' of a function f if f(x)=x, and it is interesting to study when a function must have a fixed point. Although many functions have no fixed points, there are theorems that give certain conditions under which any function must have a fixed point.
One of the most famous such theorems is the Brouwer fixed-point theorem, which says in its simplest form that any continuous function <math>f: D^2 \to D^2</math> has a fixed point. This fact has a very intuitive description: if we take a sheet of graph paper (so we can label each point with coordinates), crumple it up, and set it on top of another sheet of graph paper, then there is some point on the crumpled up sheet which is directly above the corresponding point on the second sheet! While this sounds like a simple fact, proving it rigorously relies on the techniques of [[algebraic topology]].
Another celebrated example of a fixed point theorem is known as Sharkovsky's theorem. In addition to fixed points, some functions have points for which repeating the function several times yields the same result: for example, a value of x for which cos(cos(cos(x))) = x is called a 3-periodic point of the function cosine. Sharkovsky's theorem is the surprising result that if a continuous function f has points of period 3, then it has points of all other periods! For example, consider the quadratic <math>f(x) = -\frac{5}{2} x^2 + \frac{7}{2} +1</math> has f(0) = 1, f(f(0)) = f(1) = 2, f(f(f(0))) = f(f(1))=f(2) = 0. So 0 is a point of period 3 of f(x). Sharkovsky's theorem implies that there are points of every other period. For example, there is some x such that f(f(f(f(f(f(f(f(x)))))))=x! Solving this explicitly would involve solving a polynomial of degree 14, an impossible task. But Sharkovsky's theorem guarantees that there is '''some''' value of x satisfying this.
# Periodic orbits in billiards in polyhedra. I think it should be possible to say some interesting things about this but with some real content. Possibly there's a better idea.
# Periodic orbits in billiards in polyhedra. I think it should be possible to say some interesting things about this but with some real content. Possibly there's a better idea.