Conway topograph, $\mathrm{PGL}_2(\pmb{\mathbb Z})$-dynamics and two-valued groups

V. M. Buchstaber ; A. P. Veselov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 74 (2019) no. 3, pp. 387-430

Résumé

Conway's topographic approach to binary quadratic forms and Markov triples is reviewed from the point of view of the theory of two-valued groups. This leads naturally to a new class of commutative two-valued groups, which we call involutive. It is shown that the two-valued group of Conway's lax vectors plays a special role in this class. The group $\mathrm{PGL}_2(\mathbb Z)$ describing the symmetries of the Conway topograph acts by automorphisms of this two-valued group. Binary quadratic forms are interpreted as primitive elements of the Hopf 2-algebra of functions on the Conway group. This fact is used to construct an explicit embedding of the Conway two-valued group into $\mathbb R$ and thus to introduce a total group ordering on it. The two-valued algebraic involutive groups with symmetric multiplication law are classified, and it is shown that they are all obtained by the coset construction from the addition law on elliptic curves. In particular, this explains the special role of Mordell's modification of the Markov equation and reveals its connection with two-valued groups in $K$-theory. The survey concludes with a discussion of the role of two-valued groups and the group $\mathrm{PGL}_2(\mathbb Z)$ in the context of integrability in multivalued dynamics. Bibliography: 104 titles.

Détail

Keywords: Conway topograph, modular group, two-valued groups, algebraic discrete-time dynamics, integrability.

Bibliographie

[1] N. H. Abel, “Untersuchung der Functionen zweier unabhängig veränderlichen Größen $x$ und $y$, wie $f(x, y)$, welche die Eigenschaft haben, daß $f(z, f(x, y))$ eine symmetrische Function von $z$, $x$ und $y$ ist”, J. Reine Angew. Math., 1826:1 (1826), 11–15 | DOI | MR | Zbl

[2] M. Aigner, Markov's theorem and 100 years of the uniqueness conjecture. A mathematical journey from irrational numbers to perfect matchings, Springer, Cham, 2013, x+257 pp. | DOI | MR | Zbl

[3] M. Akhtar, T. Coates, S. Galkin, A. M. Kasprzyk, “Minkowski polynomials and mutations”, SIGMA, 8 (2012), 094, 17 pp. | DOI | MR | Zbl

[4] M. E. Akhtar, A. M. Kasprzyk, “Mutations of fake weighted projective planes”, Proc. Edinb. Math. Soc. (2), 59:2 (2016), 271–285 | DOI | MR | Zbl

[5] A. Baragar, “Rational points on K3 surfaces in $\mathbb P^1\times \mathbb P^1\times \mathbb P^1$”, Math. Ann., 305 (1996), 541–558 | DOI | MR | Zbl

[6] A. Berdon, Geometriya diskretnykh grupp, Nauka, M., 1986, 301 pp. ; A. F. Beardon, The geometry of discrete groups, Grad. Texts in Math., 91, Springer-Verlag, New York, 1983, xii+337 pp. | MR | Zbl | DOI | MR | Zbl

[7] J. H. Bruinier, G. van der Geer, G. Harder, D. Zagier, The 1-2-3 of modular forms, Lectures at a summer school in Nordfjordeid, Norway (Nordfjordeid, 2004), Universitext, Springer-Verlag, Berlin, 2008, x+266 pp. | DOI | MR | Zbl

[8] J. Bourgain, A. Gamburd, P. Sarnak, Markoff surfaces and strong approximation: 1, 2016, 24 pp., arXiv: 1607.01530

[9] V. M. Bukhshtaber, “Dvuznachnye formalnye gruppy. Nekotorye prilozheniya k kobordizmam”, UMN, 26:3(159) (1971), 195–196 | MR | Zbl

[10] V. M. Bukhshtaber, “Klassifikatsiya dvuznachnykh formalnykh grupp”, UMN, 28:3(171) (1973), 173–174 | MR | Zbl

[11] V. M. Bukhshtaber, “Dvuznachnye formalnye gruppy. Algebraicheskaya teoriya i prilozheniya k kobordizmam. I”, Izv. AN SSSR. Ser. matem., 39:5 (1975), 1044–1064 | MR | Zbl

[12] V. M. Bukhshtaber, “Kharakteristicheskie klassy v kobordizmakh i topologicheskie prilozheniya teorii odnoznachnykh i dvuznachnykh formalnykh grupp”, Itogi nauki i tekhn. Ser. Sovrem. probl. matem., 10, VINITI, M., 1978, 5–178 ; V. M. Buchstaber, “Characteristic cobordism classes and topological applications of the theories of one-valued and two-valued formal groups”, J. Soviet Math., 11:6 (1979), 815–921 | MR | Zbl | DOI

[13] V. M. Bukhshtaber, “Funktsionalnye uravneniya, assotsiirovannye s teoremami slozheniya dlya ellipticheskikh funktsii, i dvuznachnye algebraicheskie gruppy”, UMN, 45:3(273) (1990), 185–186 | MR | Zbl

[14] V. M. Bukhshtaber, “Otobrazheniya Yanga–Bakstera”, UMN, 53:6(324) (1998), 241–242 | DOI | MR | Zbl

[15] V. M. Buchstaber, “$n$-valued groups: theory and applications”, Mosc. Math. J., 6:1 (2006), 57–84 | DOI | MR | Zbl

[16] V. M. Buchstaber, V. I. Dragović, “Two-valued groups, Kummer varieties, and integrable billiards”, Arnold Math. J., 4:1 (2018), 27–57 | DOI | MR | Zbl

[17] V. M. Bukhshtaber, A. N. Kholodov, “Topologicheskie konstruktsii, svyazannye s mnogoznachnymi formalnymi gruppami”, Izv. AN SSSR. Ser. matem., 46:1 (1982), 3–27 ; V. M. Bukhshtaber, A. N. Kholodov, “Topological constructions connected with many-valued formal groups”, Math. USSR-Izv., 20:1 (1983), 1–25 | MR | Zbl | DOI

[18] V. M. Bukhshtaber, D. V. Leikin, “Zakony slozheniya na yakobianakh ploskikh algebraicheskikh krivykh”, Nelineinaya dinamika, Sbornik statei, Tr. MIAN, 251, Nauka, MAIK “Nauka/Interperiodika”, M., 2005, 54–126 | MR | Zbl

[19] V. M. Bukhshtaber, A. S. Mischenko, S. P. Novikov, “Formalnye gruppy i ikh rol v apparate algebraicheskoi topologii”, UMN, 26:2(158) (1971), 131–154 | MR | Zbl

[20] V. M. Bukhshtaber, S. P. Novikov, “Formalnye gruppy, stepennye sistemy i operatory Adamsa”, Matem. sb., 84(126):1 (1971), 81–118 | MR | Zbl

[21] V. M. Buchstaber, T. E. Panov, Toric topology, Math. Surveys Monogr., 204, Amer. Math. Soc., Providence, RI, 2015, xiv+518 pp. | DOI | MR | Zbl

[22] V. M. Bukhshtaber, E. G. Ris, “Mnogoznachnye gruppy i $n$-algebry Khopfa”, UMN, 51:4(310) (1996), 149–150 | DOI | MR | Zbl

[23] V. M. Buchstaber, E. G. Rees, “Multivalued groups, their representations and Hopf algebras”, Transform. Groups, 2:4 (1997), 325–349 | DOI | MR | Zbl

[24] V. M. Buchstaber, E. G. Rees, “Multivalued groups, $n$-Hopf algebras and $n$-ring homomorphisms”, Lie groups and Lie algebras, Math. Appl., 433, Kluwer Acad. Publ., Dordrecht, 1998, 85–107 | MR | Zbl

[25] V. M. Bukhshtaber, E. G. Ris, “Koltsa nepreryvnykh funktsii, simmetricheskie proizvedeniya i algebry Frobeniusa”, UMN, 59:1(355) (2004), 125–144 | DOI | MR | Zbl

[26] V. M. Bukhshtaber, A. M. Vershik, S. A. Evdokimov, I. N. Ponomarenko, “Kombinatornye algebry i mnogoznachnye involyutivnye gruppy”, Funkts. analiz i ego pril., 30:3 (1996), 12–18 | DOI | MR | Zbl

[27] V. M. Buchstaber, A. P. Veselov, “Integrable correspondences and algebraic representations of multivalued groups”, Int. Math. Res. Not. IMRN, 1996:8 (1996), 381–400 | DOI | MR | Zbl

[28] S. Cantat, F. Loray, “Dynamics on character varieties and Malgrange irreducibility of Painlevé VI equation”, Ann. Inst. Fourier (Grenoble), 59:7 (2009), 2927–2978 | DOI | MR | Zbl

[29] A. Cayley, “A memoir on cubic surfaces”, Philos. Trans. Royal Soc. London, 159 (1869), 231–326 | DOI | Zbl

[30] A. Clay, D. Rolfsen, Ordered groups and topology, Grad. Stud. Math., 176, Amer. Math. Soc., Providence, RI, 2016, x+154 pp. ; 2015, 53 pp., arXiv: 1511.05088 | MR | Zbl

[31] H. Cohn, “Approach to Markoff's minimal forms through modular functions”, Ann. of Math. (2), 61 (1955), 1–12 | DOI | MR | Zbl

[32] Dzh. Konvei, Kvadratichnye formy, dannye nam v oschuscheniyakh, MTsNMO, M., 2008, 144 pp.; J. H. Conway, The sensual (quadratic) form, Carus Math. Monogr., 26, Math. Assoc. America, Washington, DC, 1997, xiv+152 pp. | MR | Zbl

[33] T. W. Cusick, M. E. Flahive, The Markoff and Lagrange spectra, Math. Surveys Monogr., 30, Amer. Math. Soc., Providence, RI, 1989, x+97 pp. | DOI | MR | Zbl

[34] G. Darboux, Principes de géométrie analytique, Gauthier-Villars, Paris, 1917, vi+520 pp. | MR | Zbl

[35] B. N. Delone, Peterburgskaya shkola teorii chisel, Izd-vo AN SSSR, M.–L., 1947, 421 pp. ; B. N. Delone, The St. Petersburg school of number theory, Hist. Math., 26, Amer. Math. Soc., Providence, RI, 2005, xvi+278 pp. | Zbl | DOI | MR | Zbl

[36] J. Delsarte, “Hypergroupes et opérateurs de permutation et de transmutation”, La théorie des équations aux dérivées partielles (Nancy, 1956), Colloques Internationaux du Centre National de la Recherche Scientifique, 71, CNRS, Paris, 1956, 29–45 | MR | Zbl

[37] V. G. Drinfeld, “On some unsolved problems in quantum group theory”, Quantum groups (Leningrad, 1990), Lecture Notes in Math., 1510, Springer, Berlin, 1992, 1–8 | DOI | MR | Zbl

[38] B. Dubrovin, “Geometry of 2D topological field theories”, Integrable systems and quantum groups (Montecatini Terme, 1993), Lecture Notes in Math., 1620, Fond. CIME/CIME Found. Subser., Springer, Berlin, 1996, 120–348 | DOI | MR | Zbl

[39] M. Kh. El-Khuti, “Kubicheskie poverkhnosti markovskogo tipa”, Matem. sb., 93(135):3 (1974), 331–346 | MR | Zbl

[40] A. Eremenko, A. Gabrielov, A. Hinkkanen, “Exceptional solutions to the Painlevé VI equation”, J. Math. Phys., 58:1 (2017), 012701, 8 pp. | DOI | MR | Zbl

[41] A. Felikson and P. Tumarkin, “Geometry of mutation classes of rank 3 quivers”, Arnold Math. J., 5:1 (2019), 37–55 ; (2017 (v1 – 2016)), 27 pp., arXiv: 1609.08828 | DOI | MR

[42] V. V. Fock, Dual Teichmüller spaces, 1998 (v1 – 1997), 32 pp., arXiv: dg-ga/9702018v3

[43] R. Fricke, “Ueber die Theorie der automorphen Modulgruppen”, Nachr. Ges. Wiss. Göttingen, 1896 (1896), 91–101 | Zbl

[44] G. Frobenius, “Über Gruppencharachtere”, Sitzungsber. Preuß. Akad. Wiss. Berlin, 1896 (1896), 985–1021 | Zbl

[45] F. G. Frobenius, “Über die Markoffschen Zahlen”, Sitzungsber. Preuß. Akad. Wiss. Berlin, 1913 (1913), 458–487 ; Gesammelte Abhandlungen, v. 3, Springer-Verlag, Berlin–New York, 1968 | Zbl | MR | Zbl

[46] A. A. Gaifullin, “Izgibaemye kross-politopy v prostranstvakh postoyannoi krivizny”, Algebraicheskaya topologiya, vypuklye mnogogranniki i smezhnye voprosy, Sbornik statei. K 70-letiyu so dnya rozhdeniya chlena-korrespondenta RAN Viktora Matveevicha Bukhshtabera, Tr. MIAN, 286, MAIK “Nauka/Interperiodika”, M., 2014, 88–128 ; A. A. Gaifullin, “Flexible cross-polytopes in spaces of constant curvature”, Proc. Steklov Inst. Math., 286 (2014), 77–113 ; (2014 (v1 – 2013)), 38 pp., arXiv: 1312.7608 | DOI | MR | Zbl | DOI

[47] W. M. Goldman, “The modular group action on real $SL(2)$-characters of a one-holed torus”, Geom. Topol., 7 (2003), 443–486 | DOI | MR | Zbl

[48] D. S. Gorshkov, Geometriya Lobachevskogo v svyazi s nekotorymi voprosami arifmetiki, Diss. ... kand. fiz.-matem. nauk, LGU, L., 1953, 71 pp.

[49] D. S. Gorshkov, “Geometrii Lobachevskogo v svyazi s nekotorymi voprosami arifmetiki”, Issledovaniya po teorii chisel. 4, Zap. nauch. sem. LOMI, 67, Izd-vo “Nauka”, Leningrad. otd., L., 1977, 39–85 ; D. S. Gorshkov, “Geometry of Lobachevskii in connection with certain questions of arithmetic”, J. Soviet Math., 16:1 (1981), 788–820 | MR | Zbl | DOI

[50] P. G. Grinevich, S. P. Novikov, “Strunnoe uravnenie – II. Fizicheskoe reshenie”, Algebra i analiz, 6:3 (1994), 118–140 ; P. G. Grinevich, S. P. Novikov, “String equation. II. Physical solution”, St. Petersburg Math. J., 6:3 (1995), 553–574 | MR | Zbl

[51] A. Haas, “Diophantine approximation on hyperbolic Riemann surfaces”, Acta Math., 156 (1986), 33–82 | DOI | MR | Zbl

[52] P. Hacking, Y. Prokhorov, “Smoothable del Pezzo surfaces with quotient singularities”, Compos. Math., 146:1 (2010), 169–192 | DOI | MR | Zbl

[53] R. G. Halburd, “Diophantine integrability”, J. Phys. A, 38:16 (2005), L263–L269 | DOI | MR | Zbl

[54] A. Hatcher, Topology of numbers, 2019 (v1 – 2017), 211 pp.,\par https://pi.math.cornell.edu/~hatcher/TN/TNpage.html

[55] A. Hanany, Y.-H. He, C. Sun, S. Sypsas, “Superconformal block quivers, duality trees and Diophantine equations”, J. High Energy Phys., 2013, no. 11, 017, 41 pp. | DOI

[56] F. Hirzebruch, “The signature theorem: reminiscences and recreation”, Prospects in mathematics (Princeton Univ., Princeton, NJ, 1970), Ann. of Math. Studies, 70, Princeton Univ. Press, Princeton, NJ, 1971, 3–31 | MR | Zbl

[57] A. N. W. Hone, “Diophantine non-integrability of a third-order recurrence with the Laurent property”, J. Phys. A, 39:12 (2006), L171–L177 | DOI | MR | Zbl

[58] A. Hurwitz, “Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche”, Math. Ann., 39:2 (1891), 279–284 | DOI | MR | Zbl

[59] K. Iwasaki, “A modular group action on cubic surfaces and the monodromy of the Painlevé VI equation”, Proc. Japan Acad. Ser. A Math. Sci., 78:7 (2002), 131–135 | DOI | MR | Zbl

[60] K. Iwasaki, T. Uehara, “An ergodic study of Painlevé VI”, Math. Ann., 338:2 (2007), 295–345 | DOI | MR | Zbl

[61] I. V. Izmestiev, Deformation of quadrilaterals and addition on elliptic curves, 2015, 39 pp., arXiv: 1501.07157

[62] I. Izmestiev, “Classification of flexible Kokotsakis polyhedra with quadrangular base”, Int. Math. Res. Not. IMRN, 2017:3 (2017), 715–808 | DOI | MR | Zbl

[63] B. V. Karpov, D. Yu. Nogin, “Trekhblochnye isklyuchitelnye nabory na poverkhnostyakh del Petstso”, Izv. RAN. Ser. matem., 62:3 (1998), 3–38 | DOI | MR | Zbl

[64] A. N. Kholodov, “Algebraicheskaya teoriya mnogoznachnykh formalnykh grupp”, Matem. sb., 114(156):2 (1981), 299–321 ; A. N. Holodov, “Algebraic theory of multi-valued formal groups”, Math. USSR-Sb., 42:2 (1982), 265–285 | MR | Zbl | DOI

[65] A. N. Kholodov, “Mnogomernye dvuznachnye kommutativnye formalnye gruppy”, UMN, 43:1(259) (1988), 213–214 | MR | Zbl

[66] F. Klein, Ausgewählte Kapitel der Zahlentheorie I, Vorlesung, gehalten im Wintersemester 1895/96, Göttingen, 1896, v+391 pp. | Zbl

[67] M. Kontsevich, Yu. Manin, “Gromov–Witten classes, quantum cohomology, and enumerative geometry”, Comm. Math. Phys., 164:3 (1994), 525–562 | DOI | MR | Zbl

[68] I. M. Krichever, “Ellipticheskie resheniya uravneniya Kadomtseva–Petviashvili i integriruemye sistemy chastits”, Funkts. analiz i ego pril., 14:4 (1980), 45–54 | MR | Zbl

[69] B. M. Levitan, Teoriya operatorov obobschennogo sdviga, Nauka, M., 1973, 312 pp. | MR | Zbl

[70] O. Lisovyy, Y. Tykhyy, “Algebraic solutions of the sixth Painlevé equation”, J. Geom. Phys., 85 (2014), 124–163 | DOI | MR | Zbl

[71] G. L. Litvinov, “Gipergruppy i gipergruppovye algebry”, Itogi nauki i tekhn. Ser. Sovrem. probl. matem. Nov. dostizh., 26, VINITI, M., 1985, 57–106 ; G. L. Litvinov, “Hypergroups and hypergroup algebras”, J. Soviet Math., 38:2 (1987), 1734–1761 | MR | Zbl | DOI

[72] A. V. Malyshev, “Spektry Markova i Lagranzha (obzor literatury)”, Issledovaniya po teorii chisel. 4, Zap. nauch. sem. LOMI, 67, Izd-vo “Nauka”, Leningrad. otd., L., 1977, 5–38 ; A. V. Malyshev, “Markov and Lagrange spectra (survey of the literature)”, J. Soviet Math., 16:1 (1981), 767–788 | MR | Zbl | DOI

[73] Yu. I. Manin, “Sixth Painlevé equation, universal elliptic curve, and mirror of $\mathbf P^2$”, Geometry of differential equations, Amer. Math. Soc. Transl. Ser. 2, 186, Adv. Math. Sci., 39, Amer. Math. Soc., Providence, RI, 1998, 131–151 | DOI | MR | Zbl

[74] A. Markoff, “Sur les formes quadratiques binaires indéfinies”, Math. Ann., 15:3-4 (1879), 381–406 ; 17:3 (1880), 379–399 | DOI | Zbl | DOI | MR | Zbl

[75] S. Milea, C. D. Shelley, M. H. Weissman, “Arithmetic of arithmetic Coxeter groups”, Proc. Natl. Acad. Sci. USA, 116:2 (2019), 442–449 | DOI | MR

[76] L. J. Mordell, “On the integer solutions of the equation $x^2+y^2+z^2+2xyz=n$”, J. London Math. Soc., 28:4 (1953), 500–510 | DOI | MR | Zbl

[77] D. Mamford, Lektsii o teta-funktsiyakh, Mir, M., 1988, 448 pp.; D. Mumford, Tata lectures on theta, v. I, Progr. Math., 28, Birkhäuser Boston, Inc., Boston, MA, 1983, xiii+235 pp. | DOI | MR | Zbl

[78] S. P. Novikov, “Metody algebraicheskoi topologii s tochki zreniya teorii kobordizmov”, Izv. AN SSSR. Ser. matem., 31:4 (1967), 855–951 | MR | Zbl

[79] T. Panov, Ya. Veryovkin, “On the commutator subgroup of a right-angled Artin group”, J. Algebra, 521 (2019), 284–298 | DOI | MR | Zbl

[80] U. Rehmann, E. Vinberg, “On a phenomenon discovered by Heinz Helling”, Transform. Groups, 22:1 (2017), 259–265 | DOI | MR | Zbl

[81] K. H. Rosen, Elementary number theory and its applications, 3rd ed., Addison-Wesley Publ. Co., Reading, MA, 1993, xvi+547 pp. | MR | Zbl

[82] A. N. Rudakov, “Chisla Markova i isklyuchitelnye rassloeniya na $\mathbf P^2$”, Izv. AN SSSR. Ser. matem., 52:1 (1988), 100–112 ; A. N. Rudakov, “The Markov numbers and exceptional bundles on $\mathbf P^2$”, Math. USSR-Izv., 32:1 (1989), 99–112 | MR | Zbl | DOI

[83] A. N. Rudakov, “Isklyuchitelnye vektornye rassloeniya na kvadrike”, Izv. AN SSSR. Ser. matem., 52:4 (1988), 788–812 ; A. N. Rudakov, “Exceptional vector bundles on a quadric”, Math. USSR-Izv., 33:1 (1989), 115–138 | MR | Zbl | DOI

[84] E. Selling, “Ueber die binären und ternären quadratischen Formen”, J. Reine Angew. Math., 1874:77 (1874), 143–229 | DOI | MR | Zbl

[85] C. Series, “The geometry of Markoff numbers”, The Math. Intelligencer, 7:3 (1985), 20–29 | DOI | MR | Zbl

[86] J. H. Silverman, The arithmetic of dynamical systems, Grad. Texts in Math., 241, Springer, New York, 2007, x+511 pp. | DOI | MR | Zbl

[87] A. Skorobogatov, P. Swinnerton-Dyer, “$2$-descent on elliptic curves and rational points on certain Kummer surfaces”, Adv. Math., 198:2 (2005), 448–483 | DOI | MR | Zbl

[88] A. Sorrentino and A. P. Veselov, “Markov numbers, Mather's $\beta$-function and stable norm”, Nonlinearity, 32:6 (2019), 2147–2156 ; (2017), 9 pp., arXiv: 1707.03901 | DOI | MR | Zbl

[89] K. Spalding, A. P. Veselov, “Lyapunov spectrum of Markov and Euclid trees”, Nonlinearity, 30:12 (2017), 4428–4453 | DOI | MR | Zbl

[90] K. Spalding, A. P. Veselov, “Growth of values of binary quadratic forms and Conway rivers”, Bull. Lond. Math. Soc., 50:3 (2018), 513–528 | DOI | MR | Zbl

[91] K. Spalding, A. P. Veselov, “Conway river and Arnold sail”, Arnold Math. J., 4:2 (2018), 169–177 | DOI | MR

[92] K. Spalding, A. P. Veselov, “Tropical Markov dynamics and Cayley cubic”, Integrable systems and algebraic geometry, LMS Lecture Notes Series, eds. R. Donagi, T. Shaska, Cambridge Univ. Press, Cambridge (to appear); 2019 (v1 – 2017), 11 pp., arXiv: 1707.01760

[93] B. Springborn, “The hyperbolic geometry of Markov's theorem on Diophantine approximation and quadratic forms”, Enseign. Math., 63:3-4 (2017), 333–373 | DOI | MR | Zbl

[94] A. P. Veselov, “Integriruemye otobrazheniya i algebry Li”, Dokl. AN SSSR, 292:6 (1987), 1289–1291 | MR | Zbl

[95] A. P. Veselov, “What is an integrable mapping?”, What is integrability?, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1991, 251–272 | DOI | MR | Zbl

[96] A. P. Veselov, “Integriruemye otobrazheniya”, UMN, 46:5(281) (1991), 3–45 | MR | Zbl

[97] A. P. Veselov, “Growth and integrability in the dynamics of mappings”, Comm. Math. Phys., 145:1 (1992), 181–193 | DOI | MR | Zbl

[98] A. Veselov, “Yang–Baxter maps: dynamical point of view”, Combinatorial aspect of integrable systems, MSJ Mem., 17, Math. Soc. Japan, Tokyo, 2007, 145–167 | MR | Zbl

[99] A. Veselov, Yang–Baxter and braid dynamics, Talk at GADUDIS conference (Glasgow, April 1, 2009),\par http://www.newton.ac.uk/files/seminar/20090401093010159-152138.pdf

[100] C. Vinzant, “What is ...a spectrahedron?”, Notices Amer. Math. Soc., 61:5 (2014), 492–494 | DOI | MR | Zbl

[101] H. Watanabe, “Birational canonical transformations and classical solutions of the sixth Painlevé equation”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 27:3-4 (1998), 379–425 | MR | Zbl

[102] M. H. Weissman, An illustrated theory of numbers, Amer. Math. Soc., Providence, RI, 2017, xv+323 pp. | MR | Zbl

[103] E. T. Uitteker, Dzh. N. Vatson, Kurs sovremennogo analiza, v. 1, 2, 2-e izd., Fizmatgiz, M., 1963, 343 s., 516 pp. ; E. T. Whittaker, G. N. Watson, A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions, 4th ed., Cambridge Univ. Press, Cambridge, 1927, vi+608 pp. | Zbl | MR | Zbl

[104] D. Zagier, “On the number of Markoff numbers below a given bound”, Math. Comp., 39:160 (1982), 709–723 | DOI | MR | Zbl