I recently started writing up some material on finite presentation for the CRing project. There seems to be a folk “finitely presented” approach in mathematics: to prove something over a big, scary uncountable field like $\mathbb{C}$, one argues that the problem descends to some much smaller subobject, for instance a finitely generated subring of the complex numbers. It might be possible to prove using elementary methods the analog for such smaller subobjects, from which one can deduce the result for the big object.

One way to make these ideas precise is the characteristic p principle of Abraham Robinson, which I blogged about in the past when describing the model-theoretic approach to the Ax-Grothendieck theorem. Today, I want to describe a slightly different (choice-free!) argument in this vein that I learned from an article of Serre.

Theorem 1 Let ${F: \mathbb{C}^n \rightarrow \mathbb{C}^n}$ be a polynomial map with ${F \circ F = 1_{\mathbb{C}^n}}$. Then ${F}$ has a fixed point.

We can phrase this alternatively as follows. Let ${\sigma: \mathbb{C}[x_1, \dots, x_n] \rightarrow \mathbb{C}[x_1, \dots, x_n]}$ be a ${\mathbb{C}}$-involution. Then the map on the ${\mathrm{Spec}}$‘s has a fixed point (which is a closed point).

In fact, this result can be proved using directly Robinson’s principle (exercise!). The present argument, though, has more of an algebro-geometric feel to it, and it now appears in the CRing project — you can find it in the chapter currently marked “various.(more…)

Up until now, we have concentrated on a transformation ${T}$ of a fixed measure space. We now take a different approach: ${T}$ is fixed, and we look for appropriate measures (on a fixed ${\sigma}$-algebra).  First, we will show that this space is nonempty.  Then we will characterize ergodicity in terms of extreme points.

This is the first theorem we seek to prove:

Theorem 1 Let ${T: X \rightarrow X}$ be a continuous transformation of the compact metric space ${X}$. Then there exists a probability Borel measure ${\mu}$ on ${X}$ with respect to which ${T}$ is measure-preserving.

Consider the Banach space ${C(X)}$ of continuous ${f: X \rightarrow \mathbb{C}}$ and the dual ${C(X)^*}$, which, by the Riesz representation theorem, is identified with the space of (complex) Borel measures on ${X}$. The positive measures of total mass one form a compact convex subset ${P}$ of ${C(X)^*}$ in the weak* topology by Alaoglu’s theorem. Now, ${T}$ induces a transformation of ${C(X)}$: ${f \rightarrow f \circ T}$. The adjoint transformation of ${C(X)^*}$ is given by ${\mu \rightarrow T^{-1}(\mu}$, where for a measure ${\mu}$, ${T^{-1}(\mu)(E) := \mu(T^{-1}E)}$. We want to show that ${T^*}$ has a fixed point on ${P}$; then we will have proved the theorem.

There are fancier methods in functional analysis one could use, but to finish the proof we will appeal to the simple

Lemma 2 Let ${C}$ be a compact convex subset of a locally convex space ${X}$, and let ${T: C \rightarrow C}$ be the restriction of a continuous linear map on ${X}$. Then ${T}$ has a fixed point in ${C}$. (more…)

Today we will prove the fixed point theorem, which I restate here for convenience:

Theorem 1 (Elie Cartan) Let ${K}$ be a compact Lie group acting by isometries on a simply connected, complete Riemannian manifold ${M}$ of negative curvature. Then there is a common fixed point of all ${k \in K}$.

There is a Haar measure on ${K}$. In fact, we could even construct this by picking a nonzero alternating ${n}$-tensor (where ${n=\dim K}$) at ${T_e(K)}$, and choosing the corresponding ${K}$-invariant ${n}$-form on ${K}$. This yields a functional ${C(K) \rightarrow \mathbb{R}}$, which we can assume positive by choosing the orientation appropriately. This yields the Haar measure ${d \mu}$ by the Riesz representation theorem.

Now define ${J(q) := \int_K d^2(q,kp) d \mu(k).}$ This is a continuous function ${M \rightarrow \mathbb{R}}$ which has a minimum, because ${J(q)>J(p)}$ for ${q}$ outside some compact set containing ${p}$. Let the minimum occur at ${q_0}$. I claim that the minimum is unique, which will imply that it is a fixed point of ${K}$.

It can be checked that ${J}$ is continuously differentiable; indeed, let ${q_t}$ be a curve. Then ${d^2(q_t,kp)}$ can be computed as in yesterday when ${kp \neq q_t}$; when they are equal, it is still differentiable with zero derivative because of the ${d^2}$. (I am sketching things here because I don’t currently want to dive into the technical details; see Helgason’s book for them.)

So now take ${q_t}$ to be a geodesic joining the minimal point ${q_0}$ to some other point ${q_1}$. Now

$\displaystyle \frac{d}{dt} J(q_t)|_{t=0} = \int_K \frac{d}{dt} d^2(q_t, kp) |_{t=0} d\mu(k) = 0.$

Then we get

$\displaystyle \int_K d(q_0,kp) \cos \alpha d\mu(k) = 0$

where ${\alpha}$ is an appropriate angle as in yesterday’s post. When ${q_0=kp}$, this ${\alpha}$ is not well-defined, but ${d(q_0,kp)=0}$, so it is ok. Now

$\displaystyle \int_K d^2(q_1, k.p) d\mu \geq \int_K \left( d^2(q_0, kp) + d^2(q_0,q_1) - 2 d(q_0, kp) \cos \alpha \right) d \mu(k) .$

This is because of the cosine inequality. But the cosine part vanishes, so this is strictly greater than ${J(q_0)}$. In particular, since ${q_1}$ was arbitrary, ${q_0}$ was a global minimum for ${J}$—and it is thus a fixed point.