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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Topological conjugacy of linear endomorphisms
of the 2-torus

Authors: Roy Adler, Charles Tresser and Patrick A. Worfolk
Journal: Trans. Amer. Math. Soc. 349 (1997), 1633-1652
MSC (1991): Primary 58F35, 15A36, 11E16
MathSciNet review: 1407693
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Abstract: We describe two complete sets of numerical invariants of topological conjugacy for linear endomorphisms of the two-dimensional torus, i.e., continuous maps from the torus to itself which are covered by linear maps of the plane. The trace and determinant are part of both complete sets, and two candidates are proposed for a third (and last) invariant which, in both cases, can be understood from the topological point of view. One of our invariants is in fact the ideal class of the Latimer-MacDuffee-Taussky theory, reformulated in more elementary terms and interpreted as describing some topology. Merely, one has to look at how closed curves on the torus intersect their image under the endomorphism. Part of the intersection information (the intersection number counted with multiplicity) can be captured by a binary quadratic form associated to the map, so that we can use the classical theories initiated by Lagrange and Gauss. To go beyond the intersection number, and shortcut the classification theory for quadratic forms, we use the rotation number of Poincaré.

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  • 1. R. L. Adler and R. Palais, Homeomorphic conjugacy of automorphisms on the torus, Proc. Amer. Math. Soc. 16 (1965), 1222–1225. MR 0193181, 10.1090/S0002-9939-1965-0193181-8
  • 2. J. Bernoulli, Recueil pour les Astronomes (Berlin) 1 (1772), 255-284.
  • 3. Mike Boyle and David Handelman, Algebraic shift equivalence and primitive matrices, Trans. Amer. Math. Soc. 336 (1993), no. 1, 121–149. MR 1102219, 10.1090/S0002-9947-1993-1102219-4
  • 4. Robert F. Brown, The Lefschetz fixed point theorem, Scott, Foresman and Co., Glenview, Ill.-London, 1971. MR 0283793
  • 5. Duncan A. Buell, Binary quadratic forms, Springer-Verlag, New York, 1989. Classical theory and modern computations. MR 1012948
  • 6. Harvey Cohn, A classical invitation to algebraic numbers and class fields, Springer-Verlag, New York-Heidelberg, 1978. With two appendices by Olga Taussky: “Artin’s 1932 Göttingen lectures on class field theory” and “Connections between algebraic number theory and integral matrices”; Universitext. MR 506156
  • 7. Joachim Cuntz and Wolfgang Krieger, Topological Markov chains with dicyclic dimension groups, J. Reine Angew. Math. 320 (1980), 44–51. MR 592141, 10.1515/crll.1980.320.44
  • 8. H. Davenport, The higher arithmetic, 6th ed., Cambridge University Press, Cambridge, 1992. An introduction to the theory of numbers. MR 1195784
  • 9. R. Dedekind, Schreiben an Herrn Borchardt über die Theorie der elliptischen Modulfunktionen, Journ. f. reine u. angew. Mathem. 1877, Mathematische Werke I (New York), Chelsea, 1969, pp. 174-201.
  • 10. Edward G. Effros and Chao Liang Shen, Approximately finite 𝐶*-algebras and continued fractions, Indiana Univ. Math. J. 29 (1980), no. 2, 191–204. MR 563206, 10.1512/iumj.1980.29.29013
  • 11. E. Galois, Démonstration d'un théorème sur les fractions continues périodiques, Annales de Math. 1829, {\OE}vres Mathématiques d'Évariste Galois, 2nd edition (Paris), Gauthiers-Villars, 1951, pp. 1-8.
  • 12. Carl Friedrich Gauss, Disquisitiones arithmeticae, Translated into English by Arthur A. Clarke, S. J, Yale University Press, New Haven, Conn.-London, 1966. MR 0197380
  • 13. John R. Silvester, A matrix method for solving linear congruences, Math. Mag. 53 (1980), no. 2, 90–92. MR 567956, 10.2307/2689954
  • 14. G. Humbert, Sur les fractions continues ordinaires et les formes quadratiques binaires indéfinies, J. Math. Pure Appl. 2 (1916), 104-154.
  • 15. S. Katok, Coding of closed geodesics after Gauss and Morse, To appear in Geometriae Dedicata.
  • 16. Y. Katznelson and D. Ornstein, The differentiability of the conjugation of certain diffeomorphisms of the circle, Ergodic Theory Dynam. Systems 9 (1989), no. 4, 643–680. MR 1036902, 10.1017/S0143385700005277
  • 17. Felix Klein, Vorlesungen über die Theorie der elliptischen Modulfunktionen. Band II: Fortbildung und Anwendung der Theorie, Ausgearbeitet und vervollständigt von Robert Fricke. Nachdruck der ersten Auflage. Bibliotheca Mathematica Teubneriana, Band 11, Johnson Reprint Corp., New York; B. G. Teubner Verlagsgesellschaft, Stuttgart, 1966 (German). MR 0247997
  • 18. J. L. Lagrange, Additions au mémoire sur la résolution des équations numériques, Nouveaux Memoires de l'Acad. Berlin, 1770, {\OE}vres, volume 2 (Paris), Gauthiers-Villars, 1868, pp. 603-615.
  • 19. -, Recherches d'arithmétique, Nouveaux Mémoires de l'Acad. Berlin, 1773, {\OE}vres, volume 3 (Paris), Gauthiers-Villars, 1869, pp. 695-758.
  • 20. -, Recherches d'arithmétique, Nouveaux Mémoires de l'Acad. Berlin, 1775, {\OE}vres, volume 3 (Paris), Gauthiers-Villars, 1869, pp. 759-795.
  • 21. -, Additions aux Elements d'Algèbre d'Euler: Analyse Indéterminée. St. Petersburg, 1798, {\OE}vres, volume 7 (Paris), Gauthiers-Villars, 1877, pp. 5-180.
  • 22. C. G. Latimer and C. C. MacDuffee, A correspondence between classes of ideals and classes of matrices, Ann. Mathematics 34 (1933), 313-316.
  • 23. G. Lejeune-Dirichlet, Simplification de la théorie des formes binaires du second degré à déterminant positif, J. de Math. 1857, Mathematische Werke II (New York), Chelsea, 1969, pp. 159-181.
  • 24. William Judson LeVeque, Topics in number theory. Vols. 1 and 2, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1956. MR 0080682
  • 25. G. B. Mathews, Theory of numbers, 2nd ed, Chelsea Publishing Co., New York, 1961. MR 0126402
  • 26. R. A. Mollin, Quadratics, CRC Press, Boca Raton, 1995. CMP 96:11
  • 27. H. Poincaré, Sur les courbes définies pas des équations différentielles, J. Math Pures et Appl. $4^{\mbox {\`{e}me}}$ série 1 1885, {\OE}vres Complètes, t. 1 (Paris), Gauthier-Villars, Paris, 1951, pp. 90-158.
  • 28. J.-A. Serret, Développements sur une classe d'équations, J. de Math. 15 (1850), 152-168.
  • 29. H. J. S. Smith, Mémoire sur les équations modulaires, Atti ar Accad. Lincei 1877, Collected Papers, volume 2 (New York), Chelsea, 1965, pp. 224-241.
  • 30. -, Report on the theory of numbers, Part III, Report of the British Association 1861, Collected Papers, volume 1 (New York), Chelsea, 1965, pp. 163-228.
  • 31. Olga Taussky, On a theorem of Latimer and MacDuffee, Canadian J. Math. 1 (1949), 300–302. MR 0030491
  • 32. Harvey Cohn, A classical invitation to algebraic numbers and class fields, Springer-Verlag, New York-Heidelberg, 1978. With two appendices by Olga Taussky: “Artin’s 1932 Göttingen lectures on class field theory” and “Connections between algebraic number theory and integral matrices”; Universitext. MR 506156
  • 33. R. F. Williams, The “𝐷𝐴” maps of Smale and structural stability, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 329–334. MR 0264705

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Additional Information

Roy Adler
Affiliation: I.B.M., P.O. Box 218, Yorktown Heights, New York 10598

Charles Tresser

Patrick A. Worfolk
Affiliation: The Geometry Center, 1300 S. Second St., Minneapolis, Minnesota 55454

Received by editor(s): November 13, 1995
Article copyright: © Copyright 1997 American Mathematical Society