To recall notation from last time, given a matrix \(A\in \textrm{PSL}(n,\mathbb{R})\) we let \(r(A)\) denote the spectral radius, the modulus of the largest eigenvalue. We will let \(J\) denote a Jordan block of a matrix, and will work with the convention that both the diagonal and super-diagonal have the eigenvalue (as opposed to the above-diagonal 1’s, as is standard). For example:

\[J=\lambda(I+N)=\left (\begin{array}{ccc}\lambda &\lambda &0\\0&\lambda &1\\0&0&\lambda\end{array}\right)\]

We define an ordering on the Jordan blocks of a matrix by considering the associated ordered pair (under the lexiographic order) \(\textrm{power}(J)=(\vert\lambda\vert,n)\) where \(\lambda\) is the associated eigenvalue and \(n\) is the size of the block.

Our goal here is to describe the behavior of projective transformations, by looking at iterations of them and qualitatively describing the dynamics. However first a few definitions are in order. Given a topological space \(X\) and a continuous map \(f:X\to X\) we define the forward orbit of \(x\in X\) to be the set \(\{f^n(x)\}_{n\in \mathbb{N}}\). The \(\omega\)-limit set, denoted \(\omega(f,x)\) is the set of accumulation points of the forward orbit, where for a subset \(U\subset X\) we define \(\omega(f,U)=\bigcup_{x\in U}\omega(f,x)\).

For the following defintions we will be considering positive projective space \(\mathbb{S}^n=\mathbb{P}_+(\mathbb{R}^{n+1})\) together with its projective transformations \(\textrm{SL}^\pm(n+1,\mathbb{R})\). Given a such projective transformation \(A\) we may define \(E_\mathbb{C}(A)\) to be the vector space spanned by the eigenvectors with largest modulus eigenvalue, more precisely \(E_\mathbb{C}(A)=\textrm{Span}_\mathbb{C}\{v\in\mathbb{C}^n\;\mid\; Av=\lambda v,\;\; \vert\lambda\vert=r(A)\}\). Using this we then define \(E(A)=\mathbb{R}^{n+1}\cap E_\mathbb{C}(A)\) and \(\mathbb{P}(E(A))\) as its projectivization.

As a quick example, consider the map \(A=\textrm{diag}(2,2,\frac{1}{4})\in\textrm{SL}^\pm(3,\mathbb{R})\) and let \(\Delta\subset \mathbb{S}^2\) be the triangle with vertices \([e_1],[e_2]\) and \([e_3]\). Here \(\mathbb{P}(E(A))\) is the projective line \(\ell\) containing \([e_1]\) and \([e_2]\), and the forward orbit of any point in \(\Delta\) accumulates along this line; that is, \(\omega(A,\mathring{\Delta})=\ell\).

There’s a very useful proposition from linear algebra which will help us understand the dynamics of projective transformations such as the example above which we will state here quickly. Given any \(A\in\mathrm{SL}^\pm(n+1,\mathbb{R})\) there is a finite collection of (positive) projective subspaces \(H_1,\ldots H_m\subset \mathbb{S}^n\) such that if \(W\) is a subset of their complement with nonempty interior, \(\omega(A,W)\subset \mathbb{P}_+(E(A))\) has nonempty interior, and the action of \(A\) not only preserves \(\mathbb{P}_+(E(A))\) but restricts to an orthogonal action on it.

The utility of this is best illustrated by considering another example: let \(A\in \textrm{SL}^\pm(4,\mathbb{R})\) be the following matrix

\[A=\left(\begin{array}{cccc}2\cos\theta & -2\sin\theta &0&0\\2\sin\theta &2\cos\theta &0&0\\0&0&2&0\\0&0&0&\frac{1}{8}\end{array}\right)\]

The eigenvalues of \(A\) over the complex numbers are \(\{2,2e^{\pm i\theta},\frac{1}{8}\}\), and so \(r(A)=2\). Also, \(E_\mathbb{C}(A)\) is the (complex) span of the first three basis vectors, and so \(E(A)=\textrm{Span}_\mathbb{R}\{e_1,e_2,e_3\}\). The positive projectivization of \(E(A)\) is then a 2-sphere inside of the 3-sphere, the relevant subspaces are \(H_1=[e_4]\), \(H_2=[-e_4]\), and the restriction of the action of \(A\) to \(\mathbb{S}^2\) is simply

\[2\left (\begin{array}{ccc}\cos\theta &-\sin\theta &0\\\sin\theta & \cos\theta &0\\0&0&1\end{array}\right )\]

which is a rotation about the “\(z\)”-axis (the overall constant factor of \(r(A)\) doesn’t matter under projectivization). Thus we may describe the dynamics of \(A\) acting on \(\mathbb{S}^3\) as follows: there is a pair of two antipodal “repelling” fixed points \(H_1,H_2\) and relative to them an equitorial 2-sphere \(S^2\). This sphere rotates under iterations of \(A\), and any other point of \(\mathbb{S}^3\) is attracted towards it.

This linear-algebra-inspired description of the dynamics on \(\mathbb{S}^n\) can be used to help us understand projective transformations preserving a more general properly convex subset \(\Omega\). In fact; given a projective transformation \(A\in \mathrm{SL}^\pm(\Omega)\) we can say that the largest modulus of eigenvalue for \(A\) actually occurs as a real eigenvalue, and out of all Jordan blocks for eigenvalues of that modulus, the largest occurs for the real one (In terms of the notation we have been developing, this says that there exists a most powerful Jordan block for \(A\) with \(\lambda=r(A)\)). Furthermore, the fixed set for this must lie on the boundary.

To see this, let \(H\) be the union of the projective subspaces furnished for us by the linear algebra proposition above, and consider the \(\omega(A,\Omega\setminus H)\), which is then a subset of \(\mathbb{P}_+(E(A))\) by the proposition. Intersecting this with \(\bar{\Omega}\) gives a compact, convex set preserved by \(A\), and so by the Brower fixed point theorem, there is a fixed point \(x\in \mathbb{P}_+(E(A))\cap \bar{\Omega}\). Interpreting this, we have \([Ax]=[x]\), and so \(Ax=\lambda x\) for some \(\lambda>0\). But we already know that \(x\in E(A)\) which means that \(\vert\lambda\vert=r(A)\), and so the radius is in fact achieved by a real eigenvalue. Also, because we began with a non-elliptic transformation, \(A\) fixes no points in the interior of \(\Omega\) and so our fixed point \(x\) must lie on the boundary.

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