Notes of Richard Schwartz’ sixth Cambridge lecture 26-05-2017

PETs, pseudogroup actions, and renormalisation

Started by group theorist B.H. Neumann in 1959. Outer billiard around a convex polygon composes 180 degrees rotation through vertices.

Theorem 1 For a kite (a quadrilateral with one axial symmetry), there exist unbounded orbits iff the group generated by these rotations is indiscrete.

1. Square turning maps

Fix coordinates in the plane. The square turning map ${G_s}$ consists in tiling the plane by sidelength ${s}$ squares. Then turn each square by 90 degrees around its center. ${G_s}$ is discontinuous.

What does the group generated by ${G_1}$ and ${G_s}$ look like?

Especially interesting is the map ${G_1 G_s G_1 G_s}$, made of partial isometries whose linear parts are identity, i.e. translations. This is an example of a polygon exchange transformation (PET). Here, the polygons (called periodic islands) are rectangles.

For rational ${s}$, the pattern of rectangles is reminiscent of the continued fraction expansion of ${s}$. The pattern has flaws. The orbit gets longer, and fractal-looking, when the expansion gets longer. I am unable to explain this completely.

2. PETs

Interval exchange transformations arose as first return maps of measured foliations or rational billiards. A polygon exchange transformation is the same with intervals replaced by convex polygons.

2.1. Renormalization

The first return map on a polygon is often again a PET (sometimes, there are infinitely many pieces).

2.2. Compactification

This is a locally affine equivariant map of an unbounded PET into a bounded PET (i.e. a PET on a torus).

Example. An invariant line in a PET on a torus.

2.3. Construction of PETs

Given two lattices which share a common fundamental domain ${F}$, a generic point can be mapped to ${F}$ by an element of ${L_1}$

More generally, take two fundamental domains shared by two lattices. Example: the octapet, depending on one parameter ${s}$.

Theorem 2 Let ${R(s)=1-s}$ or ${frac(1/2s)}$ depending wether ${s<1/2}$ or ${s>1/2}$. If ${t=R(s)}$, then octapet${(t)}$ is a renormalization of octapet${(s)}$.

Consequences. The aperiodic set has positive codimension.

2.4. An algebraic family in ${{\mathbb C}^n}$

In ${{\mathbb C}^n}$, ${n=2,6,10,14,...}$, I define pairs of lattices associated to the ring ${{\mathbb Z}[i]}$. We call them complex octapets.

Theorem 3 Let ${e}$ be a composition of ${n}$ square turning maps for ${n=2,6,10,14,...}$. Then ${f=e^2}$ has an affine compactification from the algebraic family.

2.5. Outer billiards compactification

Theorem 4 Let ${P}$ be a polygon without parallel sides. Then the second return map to a strip admits, outside a bounded set, a double lattice compactification.

Experiments show that PETs are nearly renormalizable, but never exactly: tilings arose, but with mistakes.

3. Pseudo-group actions

Elements come with a domain. Make a graph whose vertices are points and edges join points mapped to each other by some element of the pseudo-group. The connected components are the orbits.

Now replace points by subsets. The maximal connected components are called tiles.

Example. The ${D_4}$ pseudogroup action. ${T}$ is the set of reflections in complexe coordinate hyperplanes in ${{\mathbb C}^2}$. Let ${h}$ be the involution which is a Hadamard matrix up to scale. Let ${S=T\cup hTh^{-1}}$. This pseudo-group contains the complex octapet.

Experiments show that every complex octapet tile is a union of pseudo-group tiles.

Corollary. The ${D_4}$ pseudo-group has an exact tiling by polytopes having ${D_4}$-symmetry. It is exactly renormalizable. This provides a description for the square turning map, which arises as a invariant plane in it.

The result on outer billiards requires some extra work.