The Octagonal PETs By Richard Evan Schwartz
The Octagonal PETsbyRichard Evan Schwartz1
Contents1 Introduction1.1 What is a PET? . . . . . . . . . . . .1.2 Some Examples . . . . . . . . . . . .1.3 Goals of the Monograph . . . . . . .1.4 The Octagonal PETs . . . . . . . . .1.5 The Main Theorem: Renormalization1.6 Corollaries of The Main Theorem . .1.6.1 Structure of the Tiling . . . .1.6.2 Structure of the Limit Set . .1.6.3 Hyperbolic Symmetry . . . .1.7 Polygonal Outer Billiards . . . . . . .1.8 The Alternating Grid System . . . .1.9 Computer Assists . . . . . . . . . . .1.10 Organization . . . . . . . . . . . . .2 Background2.1 Lattices and Fundamental Domains2.2 Hyperplanes . . . . . . . . . . . . .2.3 The PET Category . . . . . . . . .2.4 Periodic Tiles for PETs . . . . . . .2.5 The Limit Set . . . . . . . . . . . .2.6 Some Hyperbolic Geometry . . . .2.7 Continued Fractions . . . . . . . .2.8 Some Analysis . . . . . . . . . . . Friends of the Octagonal PETs3 Multigraph PETs3.1 The Abstract Construction . . . . .3.2 The Reflection Lemma . . . . . . .3.3 Constructing Multigraph PETs . .3.4 Planar Examples . . . . . . . . . .3.5 Three Dimensional Examples . . .3.6 Higher Dimensional Generalizations2.40.41414344454647
4 The4.14.24.34.44.54.64.7Alternating Grid SystemBasic Definitions . . . . . . . .Compactifying the Generators .The PET Structure . . . . . . .Characterizing the PET . . . .A More Symmetric Picture . . .4.5.1 Canonical Coordinates .4.5.2 The Double Foliation . .4.5.3 The Octagonal PETs . .Unbounded Orbits . . . . . . .The Complex Octagonal PETs .4.7.1 Complex Coordinates . .4.7.2 Basic Features . . . . . .4.7.3 Additional Symmetry . .48485052545555565757585858595 Outer Billiards on Semiregular Octagons5.1 The Basic Sets . . . . . . . . . . . . . .5.2 The Far Partition . . . . . . . . . . . . .5.3 The First Return Map . . . . . . . . . .5.4 The Necklace Orbits . . . . . . . . . . .5.5 Parallelograms, Halfbones, and Dogbones5.6 The Dogbone Map . . . . . . . . . . . .5.7 The First Conjugacy . . . . . . . . . . .5.8 The Second Conjugacy . . . . . . . . . .6060626365666870746 Quarter Turn Compositions6.1 Basic Definitions . . . . . . . .6.2 The Polytope Graph . . . . . .6.3 QTCs and Polygon Graphs . . .6.4 QTCs and Outer Billiards . . .6.5 QTCs and Double Lattice PETs.757576788082II.Renormalization and Symmetry847 Elementary Properties857.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857.2 Intersection of the Parallelograms . . . . . . . . . . . . . . . . 863
7.37.47.57.67.77.8Intersection of the LatticesRotational Symmetry . . .Central Tiles . . . . . . .Inversion Symmetry . . . .Insertion Symmetry . . . .The Tiling in Trivial Cases.8686878888898 Orbit Stability and Combinatorics8.1 A Bound on Coefficients . . . . .8.2 Sharpness . . . . . . . . . . . . .8.3 The Arithmetic Graph . . . . . .8.4 Orbit Stability . . . . . . . . . . .8.5 Uniqueness and Convergence . . .8.6 Ruling out Thin Regions . . . . .8.7 Joint Convergence . . . . . . . . .90909192939495969 Bilateral Symmetry989.1 Pictures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989.2 Definitions and Formulas . . . . . . . . . . . . . . . . . . . . . 10010 Proof of the Main Theorem10310.1 Discussion and Overview . . . . . . . . . . . . . . . . . . . . . 10310.2 Proof of Lemma 10.5 . . . . . . . . . . . . . . . . . . . . . . . 10510.3 Proof of Lemma 10.6 . . . . . . . . . . . . . . . . . . . . . . . 10711 The11.111.211.311.411.511.611.7Renormalization MapElementary Properties . . . . . . . . . . . . .The Even Expanson . . . . . . . . . . . . . .Oddly Even Numbers . . . . . . . . . . . . . .The Even Expansion and Continued FractionsDiophantine Approximation . . . . . . . . . .Dense Orbits . . . . . . . . . . . . . . . . . .Proof of the Triangle Lemma . . . . . . . . . .109. 109. 110. 111. 112. 113. 114. 11512 Properties of the Tiling11712.1 Tedious Special Cases . . . . . . . . . . . . . . . . . . . . . . . 11712.2 Classification of Tile Shapes . . . . . . . . . . . . . . . . . . . 11812.3 Classification of Stable Orbits . . . . . . . . . . . . . . . . . . 1204
12.4 Existence of Square Tiles . . . . . . . . . . . . . . . . . . . . . 12112.5 The Oddly Even Case . . . . . . . . . . . . . . . . . . . . . . 12312.6 Density of Shapes . . . . . . . . . . . . . . . . . . . . . . . . . 123IIIMetric Properties13 The13.113.213.3124Filling Lemma125The Layering Constant . . . . . . . . . . . . . . . . . . . . . . 125The Filling Lemma, Part 1 . . . . . . . . . . . . . . . . . . . . 126The Filling Lemma, Part 2 . . . . . . . . . . . . . . . . . . . . 12914 The Covering Lemma13014.1 The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . 13014.2 Some Additional Pictures . . . . . . . . . . . . . . . . . . . . 13515 Further Geometric Results13615.1 The Area Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 13615.2 Tiles in Symmetric Pieces . . . . . . . . . . . . . . . . . . . . 13715.3 Pyramids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13916 Properties of the Limit Set16.1 Elementary Topological Properties16.2 Zero Area . . . . . . . . . . . . . .16.3 Projections of the Limit Set . . . .16.4 Finite Unions of Lines . . . . . . .16.5 Existence of Aperiodic Points . . .16.6 Hyperbolic Symmetry . . . . . . .140. 140. 141. 142. 144. 145. 14617 Hausdorff Convergence17.1 Results about Patches . . . . . . . . . . . . . . . . . . . . .17.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . .17.3 Covering . . . . . . . . . . . . . . . . . . . . . . . . . . . . .147. 147. 148. 15018 Recurrence Relations.15419 Hausdorff Dimension Bounds15919.1 The Upper Bound Formula . . . . . . . . . . . . . . . . . . . . 15919.2 A Formula in the Oddly Even Case . . . . . . . . . . . . . . . 1605
19.319.419.519.6IVOne Dimensional ExamplesA Warm-Up Case . . . . . .Most of The General BoundDealing with the Exceptions.Topological Properties.16116216316716920 Controlling the Limit Set20.1 The Shield Lemma . . . . . . . . . .20.2 Another Version of the Shield Lemma20.3 The Pinching Lemma . . . . . . . . .20.3.1 Case 1 . . . . . . . . . . . . .20.3.2 Case 2 . . . . . . . . . . . . .20.3.3 Case 3 . . . . . . . . . . . . .20.4 Rational Oddly Even Parameters . .181. 181. 183. 186. 187.190. 190. 191. 192. 19423 The23.123.223.3Forest CaseReduction to the Loops Theorem . . . . . . . . . . . . . . .Proof of the Loops Theorem . . . . . . . . . . . . . . . . . .An Example . . . . . . . . . . . . . . . . . . . . . . . . . . .196. 196. 197. 19824 The24.124.224.324.4Cantor Set CaseUnlikely Sets . . . . . . .Tails and Anchored PathsAcute Crosscuts . . . . . .The Main Argument . . .Arc CaseThe Easy Direction . . . . .A Criterion for Arcs . . . .Elementary Properties of theVerifying the Arc Criterion .22 Further Symmetries of the22.1 Zones . . . . . . . . . . .22.2 Symmetry of Zones . . .22.3 Intersections with Zones22.4 Folding . . . . . . . . . . . . . . . . . . . .Limit Set. . . . . .Tiling. . . . . . . . . . . . . . . . .6.170170173175176177178178.21 The21.121.221.321.4.199199200201205
24.4.1 Case 1 . . . .24.4.2 Case 2 . . . .24.4.3 Case 3 . . . .24.4.4 Case 4 . . . .24.5 Pictorial Explanation.25 Dynamics in the Arc Case25.1 The Main Result . . . . . . . .25.2 Intersection with the Partitions25.3 The Rational Case . . . . . . .25.4 Measures of Symmetric Pieces .25.5 Controlling the Measures . . . .25.6 The End of the Proof . . . . . .V.209. 209. 211. 213. 215. 216. 217Computational Details21826 Computational Methods26.1 The Fiber Bundle Picture . . .26.2 Avoiding Computational Error .26.3 Dealing with Polyhedra . . . . .26.4 Verifying the Partition . . . . .26.5 Verifying Outer Billiards Orbits26.6 A Planar Approach . . . . . . .26.7 Generating the Partitions . . .27 The Calculations27.1 Calculation 127.2 Calculation 227.3 Calculation 327.4 Calculation 427.5 Calculation 527.6 Calculation 627.7 Calculation 727.8 Calculation 827.9 Calculations 927.10Calculation 1027.11Calculation 11.205205206207207.7.219. 219. 221. 222. 224. 224. 226. 228.229. 229. 230. 231. 233. 234. 235. 235. 236. 236. 236. 237
27.12Calculation 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . 23828 The Raw Data28.1 A Guide to the Files . . . . . . . . . . .28.2 The Main Domain . . . . . . . . . . . .28.3 The Symmetric Pieces . . . . . . . . . .28.4 Period Two Tiles . . . . . . . . . . . . .28.5 The Domains from the Main Theorem .28.6 The Polyhedra in the Partition . . . . .28.7 The Action of the Map . . . . . . . . . .28.8 The Partition for Calculation 11 . . . . .28.9 The First Partition for Calculation 12 . .28.10The Second Partition for Calculation 1229 References.241. 241. 241. 242. 242. 243. 243. 245. 246. 249. 2492528
PrefacePolytope exchange transformations are higher dimensional generalizations ofinterval exchange transformations, one dimensional maps which have beenextensively and very fruitfully studied for the past 40 years or so. Polytopeexchange transformations have the added appeal that they produce intricatefractal-like tilings. At this point, the higher dimensional versions are notnearly as well understood as their 1-dimensional counterparts, and it seemsnatural to focus on such questions as finding a robust renormalization theoryfor a large class of examples.In this monograph, we introduce a general method of constructing polytope exchange transformations (PETs) in all dimensions. Our constructionis functorial in nature. One starts with a multigraph such that the verticesare labeled by convex polytopes and the edges are labeled by Euclidean lattices in such a way that each vertex label is a fundamental domain for allthe lattices labelling incident edges. There is a functor from the fundamentalgroupoid of this multigraph into the category of PETs, and the image of thisfunctor contains many interesting examples. For instance, one can producehuge multi-parameter families based on finite reflection groups.Most of the monograph is devoted to the study the simplest examples ofour construction. These examples are based on the order 8 dihedral reflectiongroup D4 . The corresponding multigraph is a digon (two vertices connectedby two edges) decorated by 2-dimensional parallelograms and lattices. Thisinput produces a 1-parameter family of polygon exchange transformationswhich we call the Octagonal PETs. One particular parameter is closelyrelated to a system studied by Adler-Kitchens-Tresser.We show that the family of octagonal PETs has a renormalization schemein which the (2, 4, ) hyperbolic reflection triangle group acts on the parameter space (by linear fractional transformations) as a renormalization symmetry group. The underlying hyperbolic geometry symmetry of the systemallows for a complete classification of the shapes of the periodic tiles and alsoa complete classification of the topology of the limit sets.We also establish a local equivalence between outer billiards on semiregular octagons and the octagonal PETs, and this gives a similarly complete description of outer billiards on semi-regular octagons. Finally, weshow how the octagonal PETs arise naturally as invariant slices of certain of4-dimensional PETs based on deformations of the E4 lattice.9
I discovered almost all the material in this monograph by computer experimentation, and then later on found rigorous proofs. Most of the proofs hereare traditional, but the proofs do rely on 12 computer calculations. Thesecalculations are described in detail in the last part of the monograph.I wrote two interactive jave programs, OctaPET and BonePET, whichillustrate essentially all the mathematics in this monograph. The reader candownload these programs from my website (as explained at the end of theintroduction) and can use them while reading the manuscript. I wrote themonograph with the intention that a serious reader would also use the programs.I would like to thank Nicolas Bedaride, Pat Hooper, Injee Jeong, JohnSmillie, and Sergei Tabachnikov for interesting conversations about topicsrelated to this work. Some of this work was carried out at ICERM in Summer2012, and most of it was carried out during my sabbatical at Oxford in 201213. I would especially like to thank All Souls College, Oxford, for providinga wonderful research environment.My sabbatical was funded from many sources. I would like to thank theNational Science Foundation, All Souls College, the Oxford Maths Institute,the Simons Foundation, the Leverhulme Trust, the Chancellor’s Professorship, and Brown University for their support during this time period.Oxford, November 201210
1Introduction1.1What is a PET?We begin by defining the main objects of study in this monograph. §2 givesmore information about what we say here.PETs: A polytope exchange transformation (or PET) is defined by a bigpolytope X which has been partitioned in two ways into small polytopes:X m[Ai i 1m[Bi .(1)i 1What we mean is that, for each i, the two polytopes Ai and Bi are translationequivalent. That is, there is some vector Vi such that Bi Ai Vi . We alwaystake the small polytopes to be convex, but sometimes X will not be convex.We define a map f : X X and its inverse f 1 : X X by the formulasf 1 (y) y Vi y int(Bi ).f (x) x Vi x int(Ai ),(2)f is not defined on points of Ai and f 1 is not defined for points in Bi .Even though f and f 1 are not everywhere defined, almost every point of xhas a well-defined forwards and backwards orbit.The Periodic Tiling: A point p X is called periodic if f n (p) p forsome n. We will establish the following well-known results in §2: If p is aperiodic point, then there is a maximal open convex polytope Up X suchthat p Up , and f, ., f n are entirely defined on Up , and every point of Upis periodic with period n. We call Up a periodic tile. We let denote theunion of periodic tiles. We call the periodic tiling.The Limit Set: When is dense in X – and this happens in the casesof interest to us here – the limit set Λ consists of those points p such thatevery neighborhood of p intersects infinitely many tiles of . Sometimes Λis called the residual set. See §2.5 for a more general definition.The Aperiodic Set: We let Λ′ Λ denote the union of points withwell-defined orbits. These orbits are necessarily aperiodic, so we call Λ′ theaperiodic set.11
1.2Some ExamplesThe simplest examples of PETs are 1-dimensional systems, known as intervalexchange transformations (IETs). Such a system is easy to produce: Partition an interval smaller intervals, then rearrange them. IETs have beenextensively studied in the past 35 years. One very early paper is [K]; seepapers [Y] and [Z] for surveys of the literature. The Rauzy induction [R]gives a satisfying renormalization theory for the family of IETs all havingthe same number of intervals in the partition.The simplest examples of higher dimensional polytope exchange transformations are products of IETs. In this case, all the polytopes are rectangularsolids. More generally, one can consider PETs (not necessarily products)in which all the polytopes are rectangular solids. In 2 dimensions, theseare called rectangle exchanges. The paper [H] establishes some foundationalresults about rectangle exchanges.The paper [AKT] gives some early examples of piecewise isometric mapswhich are not rectangle exchanges. The main example analyzed in [AKT]produces locally the same tiling as outer billiards on the regular octagon,and also the same tiling as one of the examples studied here.The papers [T2], [AE], [Go], [Low1], and [Low2] all treat a closely related set of systems with 5-fold symmetry which produce tilings by regularpentagons and/or regular decagons. The papers [AG], [LKV], [Low1], and[Low2] deal with the case of 7-fold symmetry, which is much more difficult. The difficulty comes from the fact that exp(2πi/7) is a cubic irrational,though one case with 7-fold symmetry is analyzed completely in [Low2].Outer billiards on the regular n-gon furnishes an intriguing family ofPETs. The cases n 3, 4, 6 produce regular tilings of the plane, and thecases n 5, 7, 8, 10, 12, where exp(2π/n) is a quadratic irrational, can becompletely understood in terms of renormalization. See [T2] for the casen 5, and [BC] for the other cases. There is partial information about thecase n 7, and the remaining cases are not understood at all.Some definitive theoretical work concerning the entropy of PETs is donein [GH1], [GH2], and [B]. The main results here are that such systems havezero entropy, with a suitable definition of entropy.The recent paper [Hoo] is very close in spirit to our work here. In [Hoo],the author works out a renormalization scheme for a 2-parameter family of(non-product) rectangle exchange transformations.12
1.3Goals of the MonographMultigraph PETs The first goal of this monograph is to give a generalconstruction of PETs, based on decorated multigraphs. A multigraph isa graph in which one allows multiple edges connecting different vertices.The vertices are labelled by convex polytopes and the edges are labeled byEuclidean lattices, so that a vertex is incident to an edge iff the correspondingpolytope is a fundamental domain for the corresponding lattice. Given sucha multigraph G, we choose a base vertex x and we construct a functorialhomomorphismπ1 (G, x) PET(X).(3)Here π1 (G, x) is the fundamental group of G, and PET(X) is the group ofPETs whose domain is the polytope X corresponding to x. We call theresulting systems multigraph PETs. When G is a digon–i.e. two verticesconnected by two edges, we call the system a double lattice PET . We willgive a variety of nontrivial constructions of multigraph PETs, some
I discovered almost all the material in this monograph by computer exper-imentation, and then later on found rigorous proofs. Most of the proofs here are traditional, but the proofs do rely on 12 computer calculations. These calculations a
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