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The Octagonal Pets by Richard Evan Schwartz

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The Octagonal Pets by Richard Evan Schwartz The Octagonal PETs by Richard Evan Schwartz 1 Contents 1 Introduction 11 1.1 WhatisaPET?.......................... 11 1.2 SomeExamples .......................... 12 1.3 GoalsoftheMonograph . 13 1.4 TheOctagonalPETs . 14 1.5 TheMainTheorem: Renormalization . 15 1.6 CorollariesofTheMainTheorem . 17 1.6.1 StructureoftheTiling . 17 1.6.2 StructureoftheLimitSet . 18 1.6.3 HyperbolicSymmetry . 21 1.7 PolygonalOuterBilliards. 22 1.8 TheAlternatingGridSystem . 24 1.9 ComputerAssists ......................... 26 1.10Organization ........................... 28 2 Background 29 2.1 LatticesandFundamentalDomains . 29 2.2 Hyperplanes............................ 30 2.3 ThePETCategory ........................ 31 2.4 PeriodicTilesforPETs. 32 2.5 TheLimitSet........................... 34 2.6 SomeHyperbolicGeometry . 35 2.7 ContinuedFractions . 37 2.8 SomeAnalysis........................... 39 I FriendsoftheOctagonalPETs 40 3 Multigraph PETs 41 3.1 TheAbstractConstruction. 41 3.2 TheReflectionLemma . 43 3.3 ConstructingMultigraphPETs . 44 3.4 PlanarExamples ......................... 45 3.5 ThreeDimensionalExamples . 46 3.6 HigherDimensionalGeneralizations . 47 2 4 TheAlternatingGridSystem 48 4.1 BasicDefinitions ......................... 48 4.2 CompactifyingtheGenerators . 50 4.3 ThePETStructure........................ 52 4.4 CharacterizingthePET . 54 4.5 AMoreSymmetricPicture. 55 4.5.1 CanonicalCoordinates . 55 4.5.2 TheDoubleFoliation . 56 4.5.3 TheOctagonalPETs . 57 4.6 UnboundedOrbits ........................ 57 4.7 TheComplexOctagonalPETs. 58 4.7.1 ComplexCoordinates. 58 4.7.2 BasicFeatures. 58 4.7.3 AdditionalSymmetry. 59 5 OuterBilliardsonSemiregularOctagons 60 5.1 TheBasicSets .......................... 60 5.2 TheFarPartition......................... 62 5.3 TheFirstReturnMap . 63 5.4 TheNecklaceOrbits ....................... 65 5.5 Parallelograms,Halfbones,andDogbones . 66 5.6 TheDogboneMap ........................ 68 5.7 TheFirstConjugacy . 70 5.8 TheSecondConjugacy . 74 6QuarterTurnCompositions 75 6.1 BasicDefinitions ......................... 75 6.2 ThePolytopeGraph ....................... 76 6.3 QTCsandPolygonGraphs. 78 6.4 QTCsandOuterBilliards . 80 6.5 QTCsandDoubleLatticePETs. 82 II RenormalizationandSymmetry 84 7 Elementary Properties 85 7.1 Notation.............................. 85 7.2 IntersectionoftheParallelograms . 86 3 7.3 IntersectionoftheLattices . 86 7.4 RotationalSymmetry. 86 7.5 CentralTiles ........................... 87 7.6 InversionSymmetry. 88 7.7 InsertionSymmetry. 88 7.8 TheTilinginTrivialCases. 89 8 OrbitStabilityandCombinatorics 90 8.1 ABoundonCoefficients . 90 8.2 Sharpness ............................. 91 8.3 TheArithmeticGraph . 92 8.4 OrbitStability........................... 93 8.5 UniquenessandConvergence. 94 8.6 RulingoutThinRegions . 95 8.7 JointConvergence. 96 9 Bilateral Symmetry 98 9.1 Pictures .............................. 98 9.2 DefinitionsandFormulas . .100 10ProofoftheMainTheorem 103 10.1 DiscussionandOverview . .103 10.2 ProofofLemma10.5 . .105 10.3 ProofofLemma10.6 . .107 11 The Renormalization Map 109 11.1 ElementaryProperties . .109 11.2 TheEvenExpanson . .110 11.3 OddlyEvenNumbers. .111 11.4 The Even Expansion and Continued Fractions . 112 11.5 DiophantineApproximation . 113 11.6DenseOrbits ...........................114 11.7 ProofoftheTriangleLemma. 115 12 Properties of the Tiling 117 12.1 TediousSpecialCases. .117 12.2 ClassificationofTileShapes . 118 12.3 ClassificationofStableOrbits . 120 4 12.4 ExistenceofSquareTiles. .121 12.5 TheOddlyEvenCase . .123 12.6 DensityofShapes. .123 III MetricProperties 124 13 The Filling Lemma 125 13.1 TheLayeringConstant . .125 13.2 TheFillingLemma,Part1. .126 13.3 TheFillingLemma,Part2. .129 14 The Covering Lemma 130 14.1 TheMainResult . .130 14.2 SomeAdditionalPictures . .135 15FurtherGeometricResults 136 15.1 TheAreaLemma .........................136 15.2 TilesinSymmetricPieces . .137 15.3Pyramids .............................139 16PropertiesoftheLimitSet 140 16.1 ElementaryTopologicalProperties . 140 16.2ZeroArea .............................141 16.3 ProjectionsoftheLimitSet . .142 16.4 FiniteUnionsofLines . .144 16.5 ExistenceofAperiodicPoints . 145 16.6 HyperbolicSymmetry . .146 17 Hausdorff Convergence 147 17.1 ResultsaboutPatches . .147 17.2 Convergence.. .. .. ... .. .. .. .. .. ... .. .. ..148 17.3Covering..............................150 18 Recurrence Relations 154 19HausdorffDimensionBounds 159 19.1 TheUpperBoundFormula. .159 19.2 AFormulaintheOddlyEvenCase . 160 5 19.3 OneDimensionalExamples . .161 19.4 AWarm-UpCase.........................162 19.5 MostofTheGeneralBound . .163 19.6 DealingwiththeExceptions . 167 IV TopologicalProperties 169 20 Controlling the Limit Set 170 20.1 TheShieldLemma . .170 20.2 AnotherVersionoftheShieldLemma . 173 20.3 ThePinchingLemma. .175 20.3.1 Case1 ...........................176 20.3.2 Case2 ...........................177 20.3.3 Case3 ...........................178 20.4 RationalOddlyEvenParameters . 178 21 The Arc Case 181 21.1 TheEasyDirection . .181 21.2 ACriterionforArcs . .183 21.3 ElementaryPropertiesoftheLimitSet . 186 21.4 VerifyingtheArcCriterion. 187 22FurtherSymmetriesoftheTiling 190 22.1Zones................................190 22.2 SymmetryofZones . .191 22.3 IntersectionswithZones . .192 22.4Folding...............................194 23 The Forest Case 196 23.1 ReductiontotheLoopsTheorem . 196 23.2 ProofoftheLoopsTheorem . .197 23.3AnExample............................198 24 The Cantor Set Case 199 24.1UnlikelySets ...........................199 24.2 TailsandAnchoredPaths . .200 24.3 AcuteCrosscuts. .201 24.4 TheMainArgument . .205 6 24.4.1 Case1 ...........................205 24.4.2 Case2 ...........................205 24.4.3 Case3 ...........................206 24.4.4 Case4 ...........................207 24.5 PictorialExplanation . .207 25 Dynamics in the Arc Case 209 25.1 TheMainResult . .209 25.2 IntersectionwiththePartitions . 211 25.3 TheRationalCase . .213 25.4 MeasuresofSymmetricPieces . 215 25.5 ControllingtheMeasures . .216 25.6 TheEndoftheProof. .217 V ComputationalDetails 218 26 Computational Methods 219 26.1 TheFiberBundlePicture . .219 26.2 AvoidingComputationalError. 221 26.3 DealingwithPolyhedra. .222 26.4 VerifyingthePartition . .224 26.5 VerifyingOuterBilliardsOrbits . 224 26.6 APlanarApproach . .226 26.7 GeneratingthePartitions . .228 27 The Calculations 229 27.1 Calculation1 ...........................229 27.2 Calculation2 ...........................230 27.3 Calculation3 ...........................231 27.4 Calculation4 ...........................233 27.5 Calculation5 ...........................234 27.6 Calculation6 ...........................235 27.7 Calculation7 ...........................235 27.8 Calculation8 ...........................236 27.9 Calculations9. .236 27.10Calculation10. .236 27.11Calculation11. .237 7 27.12Calculation12. .238 28 The Raw Data 241 28.1 AGuidetotheFiles . .241 28.2 TheMainDomain . .241 28.3 TheSymmetricPieces . .242 28.4 PeriodTwoTiles . .242 28.5 TheDomainsfromtheMainTheorem . 243 28.6 ThePolyhedrainthePartition . 243 28.7 TheActionoftheMap . .245 28.8 ThePartitionforCalculation11 . 246 28.9 TheFirstPartitionforCalculation12. 249 28.10The Second Partition for Calculation 12 . 249 29 References 252 8 Preface Polytope exchange transformations are higher dimensional generalizations of interval exchange transformations, one dimensional maps which have been extensively and very fruitfully studied for the past 40 years or so. Polytope exchange transformations have the added appeal that they produce intricate fractal-like tilings. At this point, the higher dimensional versions are not nearly as well understood as their 1-dimensional counterparts, and it seems natural to focus on such questions as finding a robust renormalization theory for a large class of examples. In this monograph, we introduce a general method of constructing poly- tope exchange transformations (PETs) in all dimensions. Our construction is functorial in nature. One starts with a multigraph such that the vertices are labeled by convex polytopes and the edges are labeled by Euclidean lat- tices in such a way that each vertex label is a fundamental domain for all the lattices labelling incident edges. There is a functor from the fundamental groupoid of this multigraph into the category of PETs, and the image of this functor contains many interesting examples. For instance, one can produce huge multi-parameter families based on finite reflection groups. Most of the monograph is devoted to the study the simplest examples of our construction. These examples are based on the order 8 dihedral reflection group D4. The corresponding multigraph is a digon (two vertices connected by two edges) decorated by 2-dimensional parallelograms and lattices. This input produces a 1-parameter family of polygon exchange transformations which we call the Octagonal PETs. One particular parameter is closely related to a system studied by Adler-Kitchens-Tresser. We show that the family of octagonal PETs has a renormalization scheme in which the (2, 4, ) hyperbolic reflection triangle group acts on the param- eter space (by linear∞ fractional transformations) as a renormalization sym- metry group. The underlying hyperbolic geometry symmetry of the system allows for a complete classification of the shapes of the periodic tiles and also a complete classification of the topology of the limit sets. We also establish a local equivalence between outer billiards on semi- regular octagons and the octagonal PETs, and this gives a
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