Goal. Exploit the geometry of the translation surface and the way matrices act on the saddle...
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Goal. Exploit the geometry of the translation surface and the way matrices act on the saddle connection vectors. We'll be working through this in the context of the torus and saddle connections in that setting. For the torus one intuitive way to restrict the set of slopes we're considering is to consider only those saddle connections who have slope ≤ 1 and horizontal component bounded by Q. If you know about Farey fractions there's a relationship here. Exercise 2.1. Once we've restricted the set of saddle connections to those with slope ≤ 1 and horizontal component <Q, geometrically these vectors lie in the triangle with vertices at (0.0), (Q.Q) and (Q.0), we'll want to convert them all to have the more uniform bound of horizontal component ≤ 1. Find a matrix that will take vectors in the specified triangle to a new triangle where there horizontal component is now 1. What is the new geometric description of these vectors? (The answer and further details can be found in AthreyaCheung-Torus section 4.1) Consider an ordered list of saddle connections S = {₁, 2, 3,...) in this new region. This region is sometimes referred to as the vertical strip. Exercise 2.2. Consider the family of matrices given by {h. =(², 1) | SER} the action of this family (in fact it forms a subgroup) on translation surfaces is called the horocycle flow. (This defintion is also given on the first page of AthreyaCheung-Torus). 1. Let v₁ € S be a vector with positive slope. Find the value of s such that when you apply h, to v, you get a vector with slope = 0. 2. Show that differences between slopes of consecutive saddle connections in S are preserved under the action of h₂. Main Idea. Calculating the "return time" (the value of s) of a vector under horocycle flow to a specific transversal (in this case, vectors with horizontal component ≤ 1 and slope 0) gives us a list of differences between slopes. This main idea works for all translation surfaces. In the specific case of the torus there is a nice description of this transversal. The transversal is the set of all translation surfaces (in the moduli space of the torus) with a horizontal saddle connection of length 1. In the torus case this transversal can be described geometrically as = {(a,b) ER² | a,b € (0,1], a+b>1} how we get from a point in this triangle to a surface is by using the point (a, b) to specify a matrix - (8²¹) and then applying that matrix to the vectors of the unit square. Pab== Exercise 2.3. Show that these two sets are equal 1. Given a point (a,b) Ef show that it represents a translation surface with a horizontal saddle connection of length ≤ 1. 2. Harder! Given a translation surface with a horizontal saddle connection of length <1, show that it can be represented by a point (a,b) € !! (These two exercises are the proof of Lemma 2.1 in AthreyaCheung-Torus.) It turns out that you can prove that the image of the vector (0, 1) under the Pas matrix action is always the vector with smallest slope in the vertical strip, that is the next vector that will become horizontal under the action of the horocycle flow, h.. Exercise 2.4. 1. Find the image of (0, 1) under Pab 2. Find the return time (that is the value of s) of this image under h, to the transversal. (This is part of the proof of Lemma 2.2 in AthreyaCheung-Torus.) Goal. Exploit the geometry of the translation surface and the way matrices act on the saddle connection vectors. We'll be working through this in the context of the torus and saddle connections in that setting. For the torus one intuitive way to restrict the set of slopes we're considering is to consider only those saddle connections who have slope ≤ 1 and horizontal component bounded by Q. If you know about Farey fractions there's a relationship here. Exercise 2.1. Once we've restricted the set of saddle connections to those with slope ≤ 1 and horizontal component <Q, geometrically these vectors lie in the triangle with vertices at (0.0), (Q.Q) and (Q.0), we'll want to convert them all to have the more uniform bound of horizontal component ≤ 1. Find a matrix that will take vectors in the specified triangle to a new triangle where there horizontal component is now 1. What is the new geometric description of these vectors? (The answer and further details can be found in AthreyaCheung-Torus section 4.1) Consider an ordered list of saddle connections S = {₁, 2, 3,...) in this new region. This region is sometimes referred to as the vertical strip. Exercise 2.2. Consider the family of matrices given by {h. =(², 1) | SER} the action of this family (in fact it forms a subgroup) on translation surfaces is called the horocycle flow. (This defintion is also given on the first page of AthreyaCheung-Torus). 1. Let v₁ € S be a vector with positive slope. Find the value of s such that when you apply h, to v, you get a vector with slope = 0. 2. Show that differences between slopes of consecutive saddle connections in S are preserved under the action of h₂. Main Idea. Calculating the "return time" (the value of s) of a vector under horocycle flow to a specific transversal (in this case, vectors with horizontal component ≤ 1 and slope 0) gives us a list of differences between slopes. This main idea works for all translation surfaces. In the specific case of the torus there is a nice description of this transversal. The transversal is the set of all translation surfaces (in the moduli space of the torus) with a horizontal saddle connection of length 1. In the torus case this transversal can be described geometrically as = {(a,b) ER² | a,b € (0,1], a+b>1} how we get from a point in this triangle to a surface is by using the point (a, b) to specify a matrix - (8²¹) and then applying that matrix to the vectors of the unit square. Pab== Exercise 2.3. Show that these two sets are equal 1. Given a point (a,b) Ef show that it represents a translation surface with a horizontal saddle connection of length ≤ 1. 2. Harder! Given a translation surface with a horizontal saddle connection of length <1, show that it can be represented by a point (a,b) € !! (These two exercises are the proof of Lemma 2.1 in AthreyaCheung-Torus.) It turns out that you can prove that the image of the vector (0, 1) under the Pas matrix action is always the vector with smallest slope in the vertical strip, that is the next vector that will become horizontal under the action of the horocycle flow, h.. Exercise 2.4. 1. Find the image of (0, 1) under Pab 2. Find the return time (that is the value of s) of this image under h, to the transversal. (This is part of the proof of Lemma 2.2 in AthreyaCheung-Torus.)
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