Geodesic
On partially hyperbolic symplectic automorphisms of a 6-torus
Teoretičeskaâ i matematičeskaâ fizika, Tome 212 (2022) no. 2, pp. 287-302 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study automorphisms of a $6$-torus with one-dimensional stable and unstable manifolds, and a four-dimensional center manifold. Such automorphisms are generated by integer matrices and are symplectic with respect to either the standard symplectic structure on $\mathbb{R}^6$ or a nonstandard symplectic structure generated by an integer skew-symmetric nondegenerate matrix. Such a symplectic matrix generates a partially hyperbolic automorphism of the torus if its eigenvalues are given by a pair of real numbers outside the unit circle and two pairs of conjugate complex numbers on the unit circle. The classification is determined by the topology of a foliation generated by unstable (stable) leaves of the automorphism and its action on the center manifold. There are two different cases, transitive and decomposable ones. In the first case, the foliation into unstable (stable) leaves is transitive, and in the second case, the foliation itself has a subfoliation into $2$-dimensional or $4$-dimensional tori.
Mots-clés : torus automorphism, classification.
Keywords: partial hyperbolicity, symplectic 
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K. N. Trifonov. On partially hyperbolic symplectic automorphisms of a 6-torus. Teoretičeskaâ i matematičeskaâ fizika, Tome 212 (2022) no. 2, pp. 287-302. http://geodesic.mathdoc.fr/item/TMF_2022_212_2_a8/

[1] D. V. Anosov, “Geodezicheskie potoki na zamknutykh rimanovykh mnogoobraziyakh otritsatelnoi krivizny”, Tr. MIAN SSSR, 90 (1967), 3–210 | MR | Zbl

[2] J. Franks, “Anosov diffeomorphisms on tori”, Trans. Amer. Math. Soc., 145 (1969), 117–124 | DOI | MR

[3] A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and Its Applications, 54, Cambridge Univ. Press, Cambridge, 1995 | DOI | MR | Zbl

[4] A. Manning, “There are no new Anosov diffeomorphisms on tori”, Amer. J. Math., 96:3 (1974), 422–429 | DOI | MR

[5] K. Burns, A. Wilkinson, “On the ergodicity of partially hyperbolic systems”, Ann. Math., 171:1 (2010), 451–489 | DOI | MR

[6] A. Hammerlindl, R. Potrie, “Partial hyperbolicity and classification: a survey”, Ergodic Theory Dynam. Systems, 38:2 (2018), 401–443 | DOI | MR

[7] B. Hasselblatt, Ya. Pesin, “Partially hyperbolic dynamical systems”, Handbook of Dynamical Systems, v. 1, part B, Elsevier B. V., Amsterdam, 2006, 1–55 | DOI | MR

[8] F. R. Hertz, “Stable ergodicity of certain linear automorphisms of the torus”, Ann. Math. (2), 162:1 (2005), 65–107 | DOI | MR

[9] M. I. Brin, Ya. B. Pesin, “Chastichno giperbolicheskie dinamicheskie sistemy”, Izv. AN SSSR. Ser. matem., 38:1 (1974), 170–212 | DOI | MR | Zbl

[10] M. W. Hirch, C. C. Pugh, M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Spriner, Berlin, 1977 | DOI | MR

[11] L. M. Lerman, K. N. Trifonov, “Geometry of symplectic partially hyperbolic automorphisms on 4-torus”, Dynam. Syst., 35:4 (2020), 609–624 | DOI

[12] D. V. Anosov, Ya. G. Sinai, “Nekotorye gladkie ergodicheskie sistemy”, UMN, 22:5(137) (1967), 107–172 | DOI | MR | Zbl

[13] R. Bowen, “Entropy for group endomorphisms and homogeneous spaces”, Trans. Amer. Math. Soc., 153 (1971), 401–414 | DOI | MR

[14] P. R. Halmos, “On automorphisms of compact groups”, Bull. Amer. Math. Soc., 49:8 (1943), 619–624 | DOI | MR

[15] Ya. G. Sinai, “Markovskie razbieniya i U-diffeomorfizmy”, Funkts. analiz i ego pril., 2:1 (1968), 64–89 | DOI | MR | Zbl

[16] A. Katok, S. Katok, “Higher cohomology for Abelian groups of toral automorphisms II: the partially hyperbolic case, and corrigendum”, Ergodic Theory Dynam. Systems, 25:6 (2005), 1909–1917 | DOI | MR

[17] K. Shevalle, Teoriya grupp Li, IL, M., 1958 | MR

[18] V. I. Arnold, Matematicheskie metody klassicheskoi mekhaniki, URSS, M., 2017 | MR | MR

[19] N. Burbaki, Obschaya topologiya. Osnovnye struktury, Nauka, M., 1969 | MR | Zbl

[20] A. I. Maltsev, Osnovy lineinoi algebry, Nauka, M., 1970

[21] A. B. Katok, A. Niţicǎ, Rigidity in Higner Rank Abelian Group Actions, Cambridge Univ. Press, Cambridge, 2011 | DOI | MR

[22] R. Khorn, Ch. Dzhonson, Matrichnye analiz, Mir, M., 1989 | DOI | MR | MR

[23] D. Z. Arov, “O topologicheskom podobii avtomorfizmov i sdvigov kompaktnykh kommutativnykh grupp”, UMN, 18:5(113) (1963), 133–138 | MR | Zbl