Experiments with the Markoff Surface

References

• Markoff, A. (1880). Sur les formes quadratiques binaires indéfinies. Math. Ann. 17: 379–399. doi:10.1007/BF01446234
   Google Scholar
• Series, C. (1989). Symbolic dynamics and Diophantine equations In: Dodson, M. M., Vickers, J. A. G., eds. Number Theory and Dynamical Systems, Number 134 in London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press, pp. 49–67.
   Google Scholar
• Baragar, A. (1991). The Markoff equation and equations of Hurwitz. PhD Thesis, Brown University, Providence, RI, USA.
   Google Scholar
• de Courcy-Ireland, M, Magee, M. (2018). Kesten-Mckay law for the Markoff surface mod p. arXiv:1811.00113 [math.NT].
   Google Scholar
• Bourgain, J., Gamburd, A, Sarnak, P. (2016). Markoff surfaces and strong approximation: 1. arXiv. 1607:01530.
   Google Scholar
• Meiri, C, Puder, D. (2018). The Markoff group of transformations in prime and composite moduli. Duke Math. J. 167(14):2679–2720. With an appendix by D. Carmon, arXiv:1702.08358 [math.NT]. doi:10.1215/00127094-2018-0024
   Web of Science ®Google Scholar
• Cerbu, A., Gunther, E., Magee, M, Peilen, L. (2016). The cycle structure of a Markoff automorphism over finite fields. arXiv:1610.07077v2.
   Google Scholar
• Linnik, Y. V. (1944). On the least prime in an arithmetic progression I. The basic theorem. Rec. Math. (Mat. Sbornik) N.S 15(57): 139–178.
   Google Scholar
• Linnik, Y. V. (1944). On the least prime in an arithmetic progression II. The Deuring-Heilbronn phenomenon. Rec. Math. (Mat. Sbornik) N.S. 15(57): 347–368.
   Google Scholar
• Xylouris, T. (2011). On Linnik’s constant. Acta Arith. 150(1):65–91. doi:10.4064/aa150-1-4
   Web of Science ®Google Scholar
• Titchmarsh, E. C. (1930). A divisor problem. Rend. Circ. Matem. Palermo. 54: 414–429. doi:10.1007/BF03021203
   Google Scholar
• Aigner, M. (2013). Markov’s Theorem and 100 Years of the Uniqueness Conjecture: A Mathematical Journey from Irrational Numbers to Perfect Matchings. Cham: Springer International Publishing Switzerland.
   Google Scholar
• Bombieri, E. (2007). Continued fractions and the Markoff tree. Expositiones Mathematicae. 25: 187–213. doi:10.1016/j.exmath.2006.10.002
   Web of Science ®Google Scholar
• Gurwood, C. (1976). Diophantine approximation and the Markov chain. PhD Thesis. New York University, New York, NY, USA.
   Google Scholar
• Zagier, D. (1982). On the number of Markoff numbers below a given bound. Math. Comput. 39: 709–709. doi:10.2307/2007348
   Web of Science ®Google Scholar
• Mirzakhani, M. (2016). Counting mapping class group orbits on hyperbolic surfaces. arXiv:1601.03342 [math.GT].
   Google Scholar
• McCullough, D, Wanderley, M. (2013). Nielsen equivalence of generating pairs of SL(2,q). Glasgow Math. J. 55: 481–509. doi:10.1017/S0017089512000675
   Web of Science ®Google Scholar
• Carlitz, L. (1957). The number of points on certain cubic surfaces over a finite field. Boll. Un. Mat. Ital. 12(3): 19–21.
   Google Scholar
• Dodziuk, J. (1984). Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Am. Math. Soc. 284(2): 787–794. doi:10.2307/1999107
   Web of Science ®Google Scholar
• Alon, N, Milman, V. D. (1985). λ1, isoperimetric inequalities for graphs, and superconcentrators. J. Combin. Theory. 38 (1).
   Google Scholar
• Lubotzky, A., Phillips, R, Sarnak, P. (1988). Ramanujan graphs. Combinatorica 8: 261–277. (3) doi:10.1007/BF02126799
   Web of Science ®Google Scholar
• McKay, B. D. (1981). The expected eigenvalue distribution of a large regular graph. Lin. Algebra Appl. 40: 203–216. doi:10.1016/0024-3795(81)90150-6
   Web of Science ®Google Scholar
• Kesten, H. (1959). Symmetric random walks on groups. Trans. Am. Math. Soc. 92: 336–354. doi:10.2307/1993160
   Google Scholar
• Rivin, I, Sardari, N. T. (2017). Quantum chaos on random Cayley graphs of SL2[Z/pZ]. Exp. Math. 28(3): 328–341. doi:10.1080/10586458.2017.1403982
   Web of Science ®Google Scholar
• Dubrovin, B. (1995). Geometry of 2D topological field theories In Francaviglia, M., Greco, S., eds. Integrable Systems and Quantum Groups, Number 1620 in Lecture Notes in Mathematics. Springer, Berlin, 120–348.
   Google Scholar
• Dubrovin, B, Mazzocco, M. (2000). Monodromy of certain Painlevé-VI transcendents and reflection groups. Invent. Math. 141: 55–147.
   Web of Science ®Google Scholar