By Richard Evan Schwartz

A polytope trade transformation is a (discontinuous) map from a polytope to itself that could be a translation anyplace it's outlined. The 1-dimensional examples, period trade adjustments, were studied fruitfully for a few years and feature deep connections to different parts of arithmetic, akin to Teichmuller concept. This e-book introduces a normal approach for developing polytope trade alterations in better dimensions after which reports the easiest instance of the development intimately. the best case is a 1-parameter relatives of polygon alternate modifications that seems to be heavily concerning outer billiards on semi-regular octagons. The 1-parameter kinfolk admits a whole renormalization scheme, and this constitution permits a reasonably entire research either one of the procedure and of outer billiards on semi-regular octagons. the cloth during this ebook used to be came across via machine experimentation. nevertheless, the proofs are conventional, apart from a couple of rigorous computer-assisted calculations

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We identify Π with R2 in the obvious way – just drop oﬀ the zero coordinates. 37) Fj = Π ∩ X j , Lj = Π ∩ Λj for j = 1, 2. 1. In short, we see the octagonal PET at parameter s in both slices. 13. 6. Unbounded Orbits Let s be an irrational parameter. 38) F1 = R2 ∩ χ, the domain of the octagonal PET. 14. Let Λs be the aperiodic set for the octagonal PET at parameter s. 5. 7. Suppose Fs,z has unbounded orbits for some z ∈ C. Proof: We have already remarked that Fs preserves every translate of R1 .

X2N = X be the corresponding polytopes. , L2n−1 be the corresponding lattices. Starting with a typical p0 ∈ X0 , we choose the unique lattice vector V1 ∈ L1 such that p2 = p0 + L1 ∈ X2 . We then choose the unique lattice vector V3 ∈ L3 such that p4 = p2 + L3 ∈ X4 and so on, until we reach x2n = f (p0 ). The map f is our element of PET(X). Remark: The reader might wonder why we went through so much abstraction to say something so simple. For one thing, we wanted to explain the functorial nature of our construction.

3. THE FIRST RETURN MAP 45 Proof: Say that a primary line is a line extending a side of Os . Say that a secondary line is the image of a primary line under reﬂection in one of the vertices of Os . The union of secondary lines is precisely the set of lines in the fundamental strips described above. Let B be the partition for ψ 2 . Consider ﬁrst the partition A of R2 into regions where both ψ and (ψ )−1 are completely deﬁned. The regions of A are simply the complementary regions of the primary lines.