and vertically shrinked by

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1 1. A first exmple 1.1. From infinite trnsltion surfe mp to end-periodi mp. We begin with n infinite hlf-trnsltion surfe M 0 desribed s in Figure 1 nd n ffine mp f 0 defined s follows: the surfe is horizontlly nd vertilly shrinked by 2 strethed by 5 1, then the prt to the left of the vertil dshed line is sent to the prt bove the horizontl dshed line, while the prt to the right of the vertil dshed line is rotted by π nd sent to the prt below the horizontl dshed line. Let M 1 be the double over of M 0 whih is n infinite trnsltion surfe. More preisely, onsider two opies of the polygon in figure 1: P 0 nd P 1. For eh pir of edges in figure 1 with the sme lbel, if they re in the sme diretion, we glue the pirs in P 0 nd P 1 together; if they re in the different diretion, we glue the edges in P 0 to the edges in P 1 in the opposite diretion. Let f 1 be the lift of f 0 tht sends the prt of P 0 to the left of the vertil dshed line to the prt of P 0 bove the horizontl dshed line of the sme lef. f 2 f 2 f 1 f 1 e 2 e 2 e 1 e 1 b d d b Figure 1 Now we blow up M 1 into M 3 s shown in figure 2. The blow up proeeds s follows: strt with the polygonl region in figure 1. Reple eh red dot in the boundry with short r g n, nd dd line t the top-left orner whih is divided into short segments h n. Glue ll the other edges ording to the previous prgrph, then we get n infinite (topologil) surfe M 2 with boundry whose boundry urves re g n nd h n. f 1 lifts to f 2, whih sends g n to g n+1 nd h n to h n+1. Keep the gluing reltions 1

2 2 on ll other sides. Let M 3 be the double of M 2 long its boundry, nd f 3 be diffeomorphism from M 3 to itself whih is identil to f 2 when double over of f 2. Now f 3 is n end-periodi mp (.f. [F97, CC]). This blow up proess reverses the Hndel-Miller onstrution [CC], where pir of invrint lmintions, hene n ffine mp on hlf-trnsltion surfe, is obtined from n end-periodi mp. f 2 f 2 h 3 g 3 h 2 f 1 h 3 h 2 h 0 h 1 h 1 f 1 g 2 b e 2 e 2 g 3 e 1 g 1 d e 1 g 2 d g 0 b g 1 Figure 2. This is one of the two leves of X. g i nd h i re the boundry urves. S is its double. The mp F is defined s sending the region to the left of the red line to the region bove the blue line, nd sending the region to the right of the red line to the prt below the blue line on nother lef From end-periodi mp to 3-mnifold. Let T be the mpping torus of M 3 with monodromy f 3. This is non ompt 3-mnifold, sine it is mpping torus of non ompt surfe. The mpping torus onstrution restrited to the two periodi ends (the ttrting one nd the repelling one) forms two ylindril ends of the 3 mnifold s in Figure 3,

3 lso.f. [F97]. Hene, it hs nturl omptifition T by dding two opies of losed surfe homeomorphi to the quotient of either end of M 3 by f 3. More preisely, in T, onsider the prt oming from the suspension of neighborhood of the repelling end, nd djoin n idel point to every flow line in there. The set of suh idel points is homeomorphi to the quotient of this neighborhood under the mp f 3. Similrly, one n do the sme thing with the ttrting end. In prtiulr, the boundry urves of these two periodi neighborhoods (in our se, the green dotted lines in figure 2) re identified with union of loops C nd C on these two boundry surfes. 3 Figure Approximtion by sequene of finite type surfes. Glue the two boundry surfes of T together suh tht C nd C re identified, then we get losed 3-mnifold with depth 1 folition, whose infinite leves re from the mpping torus nd the ompt lef is from the boundry surfe of T, whose dul in H 1 (N) is denoted s b. Remove smll ylindril neighborhood of the ompt lef nd glue the two ends of the infinite leves togther, we get fibertion of N on irle whih orresponds to point lose to b in PH 1 (N). In terms of the surfe, wht this gives us is mp f 4 on finite surfe M 4 bsed on the end-periodi mp. M 4 is formed by gluing the preimge of the green dotted loops under some iterte of f 3 in the ttrting end to the green dotted loops in the repelling end, nd f 4 is the sme s f 3 exept on the lst step in the ttrting end, whih, insted of being sent to the next step, it is sent to the step in the repelling end glued to it. The ext wy it is sent to the repelling end is determined by how we identify the boundries of T to form N. Here, we send h n to h 3, g n to g 3, nd e n 1 to f 1 s shown in figure 2. Vrying n we get sequene of mps on surfes of inresing genus, whih represents n rithmeti sequene in fibered one

4 4 of N. We n give flt struture on eh finite surfe mp by invrint mesure lmintions, mking them two opies of the double over of hlf trnsltion surfes. In our se, one of them is over of the following: Figure 4. The dilttion onstnt λ stisfies λ 6 λ 4 λ 3 λ = 0 And, by lultion, we know tht the dilttion of f 4 on these surfes stisfy λ n λ n 2 λ = Fibered fe nd Teihmüller polynomil. The 1-eigenspe of f 4 on H 1 (M 4 ; Q) hs rnk 3 nd is generted by the duls of the two loops formed by h i s well s the dul of the green loops s shown in figure 2. Hene, the ompt 3-mnifold hs first Betti number 4. The loops formed by h i re both preserved by f 4, hene perturbing fibertion in those two diretions orrespond to doing Dehn twists on embedded tori, whih hnges neither genus (i.e. they lie in the kernel of the Thurston norm) nor the dilttion onstnt when those loops re ollpsed. 2. Other exmples Exmple 1. Bowmn [JB] desribed suh sequene formed by Arnoux- Yooz surfes. Exmple 2 (Chmnr surfe[rc]). Strt with the Chmnr surfe (Figure 5) with the Bker s mp, where the prt to the right of the vertil dshed line is sent to the prt below the horizontl dshed line: In the polygon in Figure 5, reple eh blue dot in the boundry with short r, nd dd line t the top-left orner nd line t the bottom-right

5 5 Figure 5 orner, whih re both divided into short segments. Glue ll the other edges ording to the previous prgrph, then we get n infinite (topologil) surfe with boundry nd the ffine mp lifts to n end-periodi mp with one ontrting nd one repelling end. Do double of this infinite surfe long the boundry, nd follow the sme proedure s desribed in the previous setion, we get sequene of finite surfe mps pproximting the Bker s mp on the Chmnr surfe. Figure 6 is surfe in the sequene pproximting the Chmnr surfe. It s in strt H(2). The dilttion onstnt is pproximtely whih is the root of x 4 x 3 x 2 x + 1 = 0. The next surfe in this sequene is Figure 7. It s in strt H(1, 1). The dilttion onstnt is pproximtely whih is the root of x 5 x 4 x 3 x 2 x + 1 = 0. In generl, the dilttion onstnt is root of x n x n 1 x + 1 = 0. Exmple 3. Consider pieewise liner mp from squre to itself defined s follows: deomposes the squre horizontlly into three retngles of the sme width, streth them horizontlly by 3 nd shrink them vertilly by 3, then stk them vertilly suh tht the left-most retngle is put in

6 6 Figure 6 Figure 7 the middle, the middle retngle is put t the bottom, nd the right-most retngle is rotted by π nd put t the top. This mp indues gluing on the boundry of the squre whih mkes it into hlf-trnsltion surfe of infinite type, s shown in Figure 8. Pss to double over tht mkes it trnsltion surfe, then reple the infinite one points with line segments nd do double long them s in the previous setion, we get end-periodi mp on infinite-type surfe with one ontrting nd one repelling end. Following the sme proess s

7 7 b d b d d 1 b 1 d 1 b 1 Figure 8 desribed in the previous setion we n get sequene of finite surfe with pseudo-anosov mps, one of whih is shown in Figure 9. Figure 9. The dilttion of this pseudo-anosov mp is the lrgest root of λ 8 3λ 7 2λ 5 + 4λ 4 2λ 3 3λ + 1 = 0. By lultion, the dilttion of these sequene of pseudo-anosov mpd stisfies λ 8+2n 3λ 7+2n 2λ 5+n + 4λ 4+n 2λ 3+n 3λ + 1 = 0.

8 8 3. Aknowledgements We would lso like to thnk someone. The first uthor ws prtilly supported by the ERC Grnt Nb Referenes [CH] de Crvlho, A. nd Hll, T. (2004). Unimodl generlized pseudo-anosov mps. Geom. Topol., 8, [BRW15] Bik, H. nd Rfiqi, A. nd Wu, C. Construting pseudo-anosov mps with given dilttions. Geom. Dedit (2015), DOI /s [F97] Fenley, Srgio R. End periodi surfe homeomorphisms nd 3-mnifolds. Mthemtishe Zeitshrift (1997): [CC] Cntwell, John, nd Lwrene Conlon. Endperiodi Automorphisms of Surfes nd Folitions. rxiv preprint rxiv: (2010). [JB] Bowmn, Joshu P. The omplete fmily of Arnoux-Yooz surfes. Geometrie Dedit (2013): [RC] Chmnr, Rez. Affine utomorphism groups of surfes of infinite type. Contemporry Mthemtis 355 (2004): Mthemtishes Institut, Rheinishe Friedrih-Wilhelms-Universität Bonn, Endeniher Allee 60, Bonn, Germny E-mil ddress: Deprtment of Mthemtis, Cornell University, Mlott Hll, Ith, NY 14853, USA E-mil ddress: Deprtment of Mthemtis, Cornell University, Mlott Hll, Ith, NY 14853, USA E-mil ddress: