Iterated barycentric subdivisions and steerable semi-groups
In two dimensions, there are many different affinely natural procedures on simplices: the barycentric subdivision (defined by coning and induction on dimension), yielding 6 triangles; the truncation of corners, yielding 4 triangles. The second is better behaved for numerics (triangles do not get thin, they stay similar to each other, in fact only 2 shapes are encountered).
Question. What shapes of simplices arise in an iterated subdivision scheme?
Diaconis and McMullen show that almost every triangle produced gets thin.
1. The space of shapes
I am aiming at a different question. Shapes form a topological space. Let be the space of labelled -simplices mod scaling. Say we normalize so that volume stays equal to 1. Then is a principal homogeneous space of . An affinely natural subdivision process amounts to a subdivision of the standard simplex.
Question. Does the iteration scheme produce a dense set of shapes?
The answer is positive in 2 dimensions (Barany-Beardon-Carne 1990). I gave a positive answer in 3 and 4 dimensions, I will explain my solution. My guess is that answer should be negative for large enough.
The question is equivalent to the density of the semi-group generated by the matrices that map the standard simplex to each of the labelled tiles.
Lemma: the subgroup generated by the subdivision is either discrete or dense.
The proof is not very enlightening. It follows that the subgroup generated by the subdivision is dense.
Lemma: If contains a bounded infinite walk, then the semi-group is dense in . Indeed, a long bounded walk contains segments which are close to 1, hence an expression of inverses as elements of the semi-group up to small error.
If you label right, then all generators are infinite order elliptics.
A computer search revals some infinite order elliptic elements. They represent a positive fraction of words of length 3.
Experiments suggest existence of an infinite walk up to .
Look at horoballs in . Fix an origin .
Definition 1 A horoball is special if it contains the origin in its boundary.
A set is a strong wheel if every special horoball contains a point of in its interior.
A semi-group is steerable if the orbit contains a steering wheel.
Lemma: If is steerable, then has a bounded infinite walk.
The idea is that if a walk goes far away from , use a special horoball containing it in its boundary, pick a point there, start again.
Therefore, the point is to exhibit a steering wheel.
Lemma: Consider the Hadamard map , (geodesic polar coordinates) preserving min sectional curvature. This is distance non-decreasing. This allows to carry the problem to 9-dimensional hyperbolic space.
The wheel is a finite set of 144 points on the boundary of some ball centered at . To certify that they form a steering wheel, apply the following criterion: each of them points towards the center of a special horoball. If the convex hull of these centers contains , then the horoballs cover , hence the wheel.
What makes you think the answer should be negative in high dimensions? Because is too small a number of horospheres to cover a sphere of dimension . Experiments indicate that this phenomenon could start at .
The semi-group would be uniformly discrete?