VISUALIZING THE UNIT BALL OF THE AGY NORM

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1 VISUALIZING THE UNIT BALL OF THE AGY NORM ALEX WRIGHT 1. Abstract Avila-Gouëzel-Yoccoz defied a orm o the relative cohomology H 1 (X, Σ) of a traslatio surface (X, ω), i [AGY06, Sectio 2] ad also [AG13, Sectio 5]. (The relative cohomology H 1 (X, Σ) is a easy to uderstad vector space that is costructed i a elemetary way from the surface.) This orm plays a major role i the work of the Fields medalists Avila ad Yoccoz, ad is also related to the work of the Fields medalists McMulle ad Mirzakhai. Sice its defiitio oly recetly i 2013, we still do ot have a detailed uderstadig of this orm. For example, it is ukow if the orm o the vector space H 1 (X, Σ) cotais eough iformatio to recostruct the traslatio surface (X, ω), as is famously the case i a related situatio (the Teichmüller orm). The goal of this project is to write a computer program to draw pictures of two-dimesioal slices of the uit ball. Such a picture will be a covex set i the plae. It probably wo t be a roud circle i this case, but it may look like some sort of distorted circle. It may or may ot have corers, ad it may or may ot have flat edges. This project will empirically resolve these mysteries ad lead directly to ogoig research. A traslatio surface ca be thought of as a collectio of triagles glued together, ad H 1 (X, Σ) ca be thought of as the vector space geerated by the edges, modulo the relatios imposed by requirig that the three edges of ay triagle sum to zero. The project will aturally cosist of two mai modules (as well as a fial module that combies everythig together). The first module will fid a list of saddle coectios o the surface, which are straight lies joiig the corer of oe triagle to the corer of aother triagle (ad possibly passig through may triagles i betwee). The secod module will take that list as a iput, ad via a elemetary algorithm produce the picture of the orm ball. Mathematical prerequisites: You should be truly comfortable with the abstract vector spaces, subspaces, quotiet spaces, ad dual spaces. It is t required to kow about cohomology. You should also be truly

2 2 WRIGHT comfortable with orms o vector spaces, ad kow at least oe example that does t come from a ier product; readig the first bit of the wikipedia page o orms will fulfill this requiremet. Additioally you should have at least oe of the followig two bous prerequisites. (1) You should be comfortable with surfaces; for example, you should kow that a octago with opposite sides idetified is a geus two surface. (2) You should have some experiece programmig. Course prerequisites: Oe of Math 493 (Hoors Algebra I) or Math 395 (Hoors Aalysis I) is required. Exceptios may be provided for studets with a very strog performace i Math 296 (Hoors Mathematics II) ad a ethusiastic recommedatio from a faculty member. 2. Norms A orm is a fuctio : V R from a vector space to the real umbers satisfyig the followig axioms. (1) v 0, with equality if ad oly if v = 0. (2) cv = c v. (3) v + w v + w. Give a orm, the uit ball is defied as B 1 = {v : v 1}. It satisfies the followig properties. (1) It is bouded, closed, ad cotais 0 i its iterior. (2) It is covex, i.e. it cotais the straight lie betwee ay two of its poits. (3) It is cetrally symmetric, i.e. if it cotais v the it also cotais v. Ay set B 1 satisfyig these three properties is the uit ball for some orm. So there is truly a zoo of possibilities. If v 1, v 2 are liearly idepedet vectors, we ca plot i R 2 the set of (x, y) such that xv 1 +yv 2 has orm 1 (or orm at most 1). This allows us to visualize two dimesioal slices of the uit ball. A computer program performig this visualizatio would take as iput the two vectors v 1 ad v 2, ad output a picture of a cetrally symmetric covex blob i the plae. This blob might look like a roud circle, or a square, or a diamod, or somethig vastly more complicated. 3. A family of orms o V, ad the secod module Let V deote the dual space to V, i.e. the space of liear fuctioals. Suppose S is a set of pairs (v, l v ), where v V ad l v 0. We thik of l v as some sort of legth of the vector v, but this otio of legth

3 VISUALIZING THE UNIT BALL OF THE AGY NORM 3 does t have to be defied for all vectors i V, ad i particular does t have to come from a orm. We the attempt to defie a orm S o V as follows. A elemet φ V is a liear fuctioal φ : V R. We defie φ S = φ(v) sup. (v,l v) S l v Sometimes this gives a orm; for ow, let s just assume we ve picked S i a sufficietly itelliget way that this does ideed give a orm. Example 3.1. Suppose V = R 2 ad S = {((1, 0), 1), ((0, 1), 1)}. Let φ(x, y) = ax + by. The { φ((1, 0)) φ S = sup, 1 = sup {a, b}. } φ((0, 1)) 1 So i this case S defies a orm called the sup-orm or l -orm. Example 3.2. Suppose V = R 2 ad Let φ(x, y) = ax + by. The S = {((x, y), x 2 + y 2 ) : x, y R}. φ S = sup = sup = a 2 + b 2 φ(x, y) x2 + y 2 ax + by x2 + y 2 by the Cauchy-Schwarz iequality. So i this case S defies the usual Euclidia orm, also called the l 2 -orm. The secod module of this project will take as iput a fiite family S. It will suffice to assume V = R 2. The module will cosist of writig a computer program to draw the uit ball of S. A good algorithm to do this would be to evely sample poits o the usual circle, amely {( cos 2π k, si 2π k ) } : k = 0..., 1. For each of these poits v, compute its orm, ad plot the poit (cos k2π k2π, si (cos k2π ), si k2π ) S.

4 4 WRIGHT 4. Traslatio surfaces, ad the first module The surfaces we cosider will be defied by gluig together triagles i R 2. We will oly be allowed to glue triagles alog parallel edges of the same legth, i a way so that there is oe triagle o each side of each edge. Suppose we fix a orietatio o each edge. Cocretely, that meas that each edge is ow a vector, so we ca specify that it has a tip ad a tail. We will do this i such a way that whe we glue triagles alog a edge, the orietatios agree. Let V e deote the abstract vector space geerated by the edges. A elemet of V e looks like c i v i, where v i are edges ad c i are real umbers. Let V t deote the subspace by the followig elemets: For each triagle, order the three edges e 1, e 2, e 3 i such a way that e 1 +e 2 = e 3 as vectors i R 2. The V t is spaed by the e 1 + e 2 e 3. Note that this object is ot a vector i R 2. It is a vector i V e. We defie H 1 (X, Σ) to be V e /V t. (I case you are woderig, Σ is defied to be the subset of the surface arisig from the vertices.) Exercise 4.1. Cosider ay spaig subtree of the graph of the triagulatio of the surface. This gives rise to a subspace of V e which is isomorphic to V e /V t. We defie H 1 (X, Σ) to be the dual space H 1 (X, Σ). Note that we also have H 1 (X, Σ) = H 1 (X, Σ). A saddle coectio o the surface is defied to be a straight lie joiig oe corer of a triagle to aother, possibly goig through may triagles i betwee, but ot passig through ay other corers of triagles. There are ifiitely may saddle coectios o each traslatio surface, but oly fiitely may of legth at most L. Each saddle coectio γ gives rise to a elemet [γ] of H 1 (X, Σ), ad each saddle coectio has a legth l γ. Defie ad S = {([γ], l γ ) : γ a saddle coectio} S L = {([γ], l γ ) : γ a saddle coectio of legth at most L}. The first module of the project will take L ad a surface as its iput, ad compute S L. There is already some ope source software that does some related thigs, ad we ca either try to modify it or try to imitate some of its algorithms. This software is available o the

5 VISUALIZING THE UNIT BALL OF THE AGY NORM 5 webpage of Roe Mukamel. There is also software writte by Vicet Delecroix ad Pat Hooper available at sage-flatsurf. 5. The defiitio of the AGY orm, ad the fial module The AGY orm o H 1 (X, Σ) is defied as S, where S is as above. Sice this S is ifiite, we ca t compute it, but the secod module computes S L for each L > 0. The fial module should oly be started oce the first two modules are doe or almost doe. It will accept the iput from the user, ad iterface betwee the first two modules. It will accept as iput a surface, two liearly idepedet vectors φ 1, φ 2 of H 1 (X, Σ). There will also be parameters L > 0 ad > 0 that determie the accuracy to which we wat to draw the uit ball. (We will cosider saddle coectios of legth at most L, ad plot the uit ball by samplig evely spaced poits o the circle.) It will call the secod module to compute S L. It will the traslate S L ito the cotext of R 2, allowig us to call the first module ad draw the uit ball. Refereces [AG13] Artur Avila ad Sébastie Gouëzel, Small eigevalues of the Laplacia for algebraic measures i moduli space, ad mixig properties of the Teichmüller flow, A. of Math. (2) 178 (2013), o. 2, MR [AGY06] Artur Avila, Sébastie Gouëzel, ad Jea-Christophe Yoccoz, Expoetial mixig for the Teichmüller flow, Publ. Math. Ist. Hautes Études Sci. (2006), o. 104,

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