This chapter explores some consequences of the results in Chapter 2, especially Theorem 2.3. It suggests that Theorem 2.3 gives a way to extract information from the geometry of the low capacity lines. Section 3.2 proves that, relative to the parameter p/q, the 0th block always contains a polygon whose projection onto the X-axis has diameter at least (p + q)/2. Section 3.3 elaborates on the theme in Section 3.2 to show how to extract increasingly fine scale information about the plaid polygons. Section 3.4 explains how to augment the idea in Section 3.3 to make it more useful. Section 3.5 shows how the ideas from Section 3.3 sometimes explain why the plaid model looks similar at different rational parameters.
{"title":"Using the Model","authors":"R. Schwartz","doi":"10.2307/j.ctv5rf6tz.7","DOIUrl":"https://doi.org/10.2307/j.ctv5rf6tz.7","url":null,"abstract":"This chapter explores some consequences of the results in Chapter 2, especially Theorem 2.3. It suggests that Theorem 2.3 gives a way to extract information from the geometry of the low capacity lines. Section 3.2 proves that, relative to the parameter p/q, the 0th block always contains a polygon whose projection onto the X-axis has diameter at least (p + q)/2. Section 3.3 elaborates on the theme in Section 3.2 to show how to extract increasingly fine scale information about the plaid polygons. Section 3.4 explains how to augment the idea in Section 3.3 to make it more useful. Section 3.5 shows how the ideas from Section 3.3 sometimes explain why the plaid model looks similar at different rational parameters.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133640897","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This chapter aims to prove Theorem 1.4 and Theorem 0.3, the Plaid Master Picture Theorem. Both of these results are deduced from Theorem 8.2, which says that the union PLA of plaid polygons is generated by an explicitly defined tiling classifying space (ලA, XP). Moreover, there is a nice space X which has the individual spaces XP as rational slices. The space X has a partition into convex polytopes, and one obtains the partition of XP by intersecting the relevant slice with this partition. Section 8.2 describes the space X. Section 8.3 describes the partition of X into convex integer polytopes. The partition is called the checkerboard partition. Section 8.4 explains the classifying map ΦA : Π → XP. Section 8.5 states Theorem 8.2 and deduces Theorem 1.4 and Theorem 0.3 from it.
{"title":"The Plaid Master Picture Theorem","authors":"R. Schwartz","doi":"10.2307/j.ctv5rf6tz.12","DOIUrl":"https://doi.org/10.2307/j.ctv5rf6tz.12","url":null,"abstract":"This chapter aims to prove Theorem 1.4 and Theorem 0.3, the Plaid Master Picture Theorem. Both of these results are deduced from Theorem 8.2, which says that the union PLA of plaid polygons is generated by an explicitly defined tiling classifying space (ලA, XP). Moreover, there is a nice space X which has the individual spaces XP as rational slices. The space X has a partition into convex polytopes, and one obtains the partition of XP by intersecting the relevant slice with this partition. Section 8.2 describes the space X. Section 8.3 describes the partition of X into convex integer polytopes. The partition is called the checkerboard partition. Section 8.4 explains the classifying map ΦA : Π → XP. Section 8.5 states Theorem 8.2 and deduces Theorem 1.4 and Theorem 0.3 from it.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127646506","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This chapter proves a general compactification theorem for quarter turn compositions. It is organized as follows. Section 15.2 proves a well-known result from linear algebra which will help with the material in the following section. Section 15.3 defines the map Ψ: S → Ŝ and study the dimension of its image as a function of the parameters of Τ. Recall that Τ is a composition of shears and quarter turn maps. Section 15.4 establishes Lemma 15.6, which shows that Ψ interacts in the desired way with shears. Ψ15.5 establishes Lemma 15.7, which does the same thing for quarter turn maps. Ψ15.6 combines Lemmas 15.6 and 15.7 to prove Theorem 15.1.
{"title":"Quarter Turn Compositions and PETs","authors":"R. Schwartz","doi":"10.2307/j.ctv5rf6tz.19","DOIUrl":"https://doi.org/10.2307/j.ctv5rf6tz.19","url":null,"abstract":"This chapter proves a general compactification theorem for quarter turn compositions. It is organized as follows. Section 15.2 proves a well-known result from linear algebra which will help with the material in the following section. Section 15.3 defines the map Ψ: S → Ŝ and study the dimension of its image as a function of the parameters of Τ. Recall that Τ is a composition of shears and quarter turn maps. Section 15.4 establishes Lemma 15.6, which shows that Ψ interacts in the desired way with shears. Ψ15.5 establishes Lemma 15.7, which does the same thing for quarter turn maps. Ψ15.6 combines Lemmas 15.6 and 15.7 to prove Theorem 15.1.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114832690","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This chapter gives a proof of the Intertwining Lemma. Section 20.2 lists out the formulas for all the maps involved. Section 20.3 recalls the definition of Z* and proves Statement 3 of the Intertwining Lemma. Section 20.4 proves statements 1 and 2 of the Intertwining Lemma for a single point. Section 20.5 decomposes Z* into two smaller pieces as a prelude to giving the inductive step in the proof. Section 20.6 proves the following induction step: If the Intertwining Lemma is true for g ɛ GA then it is also true for g + dTA (0, 1). Section 20.7 explains what needs to be done to finish the proof of the Intertwining Lemma. Section 20.8 proves the Intertwining Theorem for points in ΠA corresponding to the points gn = (n + 1/2)(1 + A, 1 − A) for n = 0, 1, 2, ... which all belong to GA. This result combines with the induction step to finish the proof, as explained in Section 20.7.
{"title":"The Intertwining Lemma","authors":"R. Schwartz","doi":"10.2307/j.ctv5rf6tz.24","DOIUrl":"https://doi.org/10.2307/j.ctv5rf6tz.24","url":null,"abstract":"This chapter gives a proof of the Intertwining Lemma. Section 20.2 lists out the formulas for all the maps involved. Section 20.3 recalls the definition of Z* and proves Statement 3 of the Intertwining Lemma. Section 20.4 proves statements 1 and 2 of the Intertwining Lemma for a single point. Section 20.5 decomposes Z* into two smaller pieces as a prelude to giving the inductive step in the proof. Section 20.6 proves the following induction step: If the Intertwining Lemma is true for g ɛ GA then it is also true for g + dTA (0, 1). Section 20.7 explains what needs to be done to finish the proof of the Intertwining Lemma. Section 20.8 proves the Intertwining Theorem for points in ΠA corresponding to the points gn = (n + 1/2)(1 + A, 1 − A) for n = 0, 1, 2, ... which all belong to GA. This result combines with the induction step to finish the proof, as explained in Section 20.7.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"103 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124598682","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This chapter completes the proof of the weak and strong case of the Copy Lemma. The two cases have just about the same proof. Section 25.2 proves the first two statements of the Copy Lemma. The rest of the chapter is devoted to proving the third statement. Section 25.3 proves an easy technical lemma. Section 25.4 repackages some of the results from Section 1.5. Two sequences are assigned to each rectangle in the plane: a mass sequence and a capacity sequence. It is established that these sequences determine the structure of the plaid model inside the rectangle. Section 25.5 proves a technical result about vertical light points. The final section verifies the conditions of the Matching Criterion.
{"title":"The Weak and Strong Case","authors":"R. Schwartz","doi":"10.2307/j.ctv5rf6tz.29","DOIUrl":"https://doi.org/10.2307/j.ctv5rf6tz.29","url":null,"abstract":"This chapter completes the proof of the weak and strong case of the Copy Lemma. The two cases have just about the same proof. Section 25.2 proves the first two statements of the Copy Lemma. The rest of the chapter is devoted to proving the third statement. Section 25.3 proves an easy technical lemma. Section 25.4 repackages some of the results from Section 1.5. Two sequences are assigned to each rectangle in the plane: a mass sequence and a capacity sequence. It is established that these sequences determine the structure of the plaid model inside the rectangle. Section 25.5 proves a technical result about vertical light points. The final section verifies the conditions of the Matching Criterion.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"81 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131873975","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This chapter begins Part 4 of the monograph. The goal of this part is to prove the Orbit Equivalence Theorem and the Quasi-Isomorphism Theorem. Theorem 17.1 (Orbit Equivalence) states that there is a dynamically large subset Z ⊂ X and a map Ω: Z → Y. Section 17.2 defines Z. Section 17.3 defines Ω. Section 17.4 characterizes the image Ω(Z). Section 17.5 defines a partition of Z into small convex polytopes which have the property that all the maps in Equations 17.1 and 1 are entirely defined and projective on each polytope. This allows us to verify the properties in the Orbit Equivalence Theorem just by checking what the two relevant maps do to the vertices of the new partition. Section 17.6 puts everything together and prove the Orbit Equivalence Theorem modulo some integer computer calculations. Section 17.7 discusses the computational techniques used to carry out the calculations from Section 17.6. Section 17.8 explains the calculations.
{"title":"The Orbit Equivalence Theorem","authors":"R. Schwartz","doi":"10.2307/j.ctv5rf6tz.21","DOIUrl":"https://doi.org/10.2307/j.ctv5rf6tz.21","url":null,"abstract":"This chapter begins Part 4 of the monograph. The goal of this part is to prove the Orbit Equivalence Theorem and the Quasi-Isomorphism Theorem. Theorem 17.1 (Orbit Equivalence) states that there is a dynamically large subset Z ⊂ X and a map Ω: Z → Y. Section 17.2 defines Z. Section 17.3 defines Ω. Section 17.4 characterizes the image Ω(Z). Section 17.5 defines a partition of Z into small convex polytopes which have the property that all the maps in Equations 17.1 and 1 are entirely defined and projective on each polytope. This allows us to verify the properties in the Orbit Equivalence Theorem just by checking what the two relevant maps do to the vertices of the new partition. Section 17.6 puts everything together and prove the Orbit Equivalence Theorem modulo some integer computer calculations. Section 17.7 discusses the computational techniques used to carry out the calculations from Section 17.6. Section 17.8 explains the calculations.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129122165","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This chapter proves several results about the graph grid, GA = TA (Z2). It is organized as follows. Section 19.2 proves the Grid Geometry Lemma. Section 19.3 proves the Graph Reconstruction Lemma, a result which describes how the map ΨA interacts with GA.
本章证明了关于图网格GA = TA (Z2)的几个结果。它的组织如下。第19.2节证明了网格几何引理。第19.3节证明了图重建引理，这是一个描述地图Ψ a如何与GA交互的结果。
{"title":"Geometry of the Graph Grid","authors":"R. Schwartz","doi":"10.2307/j.ctv5rf6tz.23","DOIUrl":"https://doi.org/10.2307/j.ctv5rf6tz.23","url":null,"abstract":"This chapter proves several results about the graph grid, GA = TA (Z2). It is organized as follows. Section 19.2 proves the Grid Geometry Lemma. Section 19.3 proves the Graph Reconstruction Lemma, a result which describes how the map ΨA interacts with GA.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132048025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A plaid polygon is called N-fat if it is not contained in any strip of width N. As a related notion, a plaid polygon is called N-long if it has diameter at least N. This chapter will prove Theorem 0.8. Section 22.2 studies equidistribution properties of the plaid PET map ΦA, as a function of A. Section 22.3 uses these equidistribution properties to show that the N-fat polygons essentially appear everywhere in the planar plaid model. The result is called the Ubiquity Lemma. Section 22.4 examines how the plaid model interacts with the grid of all lines of capacity at most K. Section 22.5 uses the Rectangle Lemma on many scales to show the existence of many distinct N-fat polygons. Section 2.6 discusses some properties of continued fractions and circle rotations. Finally, Section 22.7 proves the Grid Supply Lemma.
{"title":"Existence of Many Large Orbits","authors":"R. Schwartz","doi":"10.2307/j.ctv5rf6tz.26","DOIUrl":"https://doi.org/10.2307/j.ctv5rf6tz.26","url":null,"abstract":"A plaid polygon is called N-fat if it is not contained in any strip of width N. As a related notion, a plaid polygon is called N-long if it has diameter at least N. This chapter will prove Theorem 0.8. Section 22.2 studies equidistribution properties of the plaid PET map ΦA, as a function of A. Section 22.3 uses these equidistribution properties to show that the N-fat polygons essentially appear everywhere in the planar plaid model. The result is called the Ubiquity Lemma. Section 22.4 examines how the plaid model interacts with the grid of all lines of capacity at most K. Section 22.5 uses the Rectangle Lemma on many scales to show the existence of many distinct N-fat polygons. Section 2.6 discusses some properties of continued fractions and circle rotations. Finally, Section 22.7 proves the Grid Supply Lemma.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125813137","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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