June  2012, 32(6): 1965-1976. doi: 10.3934/dcds.2012.32.1965

A profinite group invariant for hyperbolic toral automorphisms

1. 

Department of Mathematics, Brigham Young University, Provo, Utah, United States

2. 

Department of Mathematics, Instituto Superior Técnico, Univ. Tec. Lisboa, Lisboa, Portugal

Received  February 2011 Revised  July 2011 Published  February 2012

For a hyperbolic toral automorphism, we construct a profinite completion of an isomorphic copy of the homoclinic group of its right action using isomorphic copies of the periodic data of its left action. The resulting profinite group has a natural module structure over a ring determined by the right action of the hyperbolic toral automorphism. This module is an invariant of conjugacy that provides means in which to characterize when two similar hyperbolic toral automorphisms are conjugate or not. In particular, this shows for two similar hyperbolic toral automorphisms with module isomorphic left action periodic data, that the homoclinic groups of their right actions play the key role in determining whether or not they are conjugate. This gives a complete set of dynamically significant invariants for the topological classification of hyperbolic toral automorphisms.
Citation: Lennard F. Bakker, Pedro Martins Rodrigues. A profinite group invariant for hyperbolic toral automorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 1965-1976. doi: 10.3934/dcds.2012.32.1965
References:
[1]

R. Adler and R. Palais, Homeomorphic conjugacy of automorphisms on the torus,, Proc. Amer. Math. Soc., 16 (1965), 1222. doi: 10.1090/S0002-9939-1965-0193181-8. Google Scholar

[2]

R. Alder, C. Tresser and P. A. Worfolk, Topological conjugacy of linear endomorphisms of the $2$-torus,, Trans. Amer. Math. Soc., 349 (1997), 1633. doi: 10.1090/S0002-9947-97-01895-3. Google Scholar

[3]

L. F. Bakker, Rigidity of projective conjugacy of quasiperiodic flows of Koch type,, Colloq. Math., 112 (2008), 291. doi: 10.4064/cm112-2-6. Google Scholar

[4]

H. Cohen, "A Course in Computational Algebraic Number Theory,'', Graduate Texts in Mathematics, 138 (1993). Google Scholar

[5]

E. C. Dade, O. Taussky and H. Zassenhaus, On the theory of orders, in particular on the semigroup of ideal classes and genera of an order in an algebraic number field,, Math. Ann., 148 (1962), 31. doi: 10.1007/BF01438389. Google Scholar

[6]

F. Grunewald and D. Segal, Some general algorithms. I: Arithmetic groups,, Ann. Math. (2), 112 (1980), 531. Google Scholar

[7]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'', With a supplementary chapter by Katok and Leonardo Mendoza, 54 (1995). Google Scholar

[8]

A. Katok, S. Katok and K. Schmidt, Rigidity of measurable structures for $\mathbbZ^d$-actions by automorphims of a torus,, Comment. Math. Helv., 77 (2002), 718. doi: 10.1007/PL00012439. Google Scholar

[9]

G. Kolutsky, Geometric continued fractions as invariants in the topological classification of Anosov diffeomorphisms of tori,, in, 354 (2009), 99. Google Scholar

[10]

D. Lind and K. Schmidt, Homoclinic points of algebraic $\mathbb Z^d$-actions,, J. Amer. Math. Soc., 12 (1999), 953. doi: 10.1090/S0894-0347-99-00306-9. Google Scholar

[11]

P. Martins Rodrigues and J. Sousa Ramos, Algebraic results on Markov shifts of torus automorphisms,, in, (1998), 436. Google Scholar

[12]

P. Martins Rodrigues and J. Sousa Ramos, Symbolic representations and $\mathbbT^n$ automorphisms,, Int. J. of Bifurcation and Chaos Appl. Sci. Engrg., 13 (2003), 2005. doi: 10.1142/S0218127403007849. Google Scholar

[13]

P. Martins Rodrigues and J. Sousa Ramos, Bowen-Franks groups as conjugacy invariants for $\mathbbT^n$-automorphisms,, Aequationes Math., 69 (2005), 231. doi: 10.1007/s00010-004-2753-7. Google Scholar

[14]

M. Newman, "Integral Matrices,'', Pure and Applied Mathematics, 45 (1972). Google Scholar

[15]

L. Ribes and P. Zalesskii, "Profinite Groups,'', Ergebnisse der Mathematik und ihrer Grenzgebiete, 40 (2000). Google Scholar

[16]

H. P. F. Swinnerton-Dyer, "A Brief Guide to Algebraic Number Theory,'', London Mathematical Society Student Texts, 50 (2001). Google Scholar

[17]

O. Taussky, Connections between algebraic number theory and integral matrices,, an appendix in H. Cohn, (1978). Google Scholar

[18]

D. I. Wallace, Conjugacy classes of hyperbolic matrices in $SL(n,\mathbbZ)$ and ideal classes in an order,, Trans. Amer. Math. Soc., 283 (1984), 177. Google Scholar

show all references

References:
[1]

R. Adler and R. Palais, Homeomorphic conjugacy of automorphisms on the torus,, Proc. Amer. Math. Soc., 16 (1965), 1222. doi: 10.1090/S0002-9939-1965-0193181-8. Google Scholar

[2]

R. Alder, C. Tresser and P. A. Worfolk, Topological conjugacy of linear endomorphisms of the $2$-torus,, Trans. Amer. Math. Soc., 349 (1997), 1633. doi: 10.1090/S0002-9947-97-01895-3. Google Scholar

[3]

L. F. Bakker, Rigidity of projective conjugacy of quasiperiodic flows of Koch type,, Colloq. Math., 112 (2008), 291. doi: 10.4064/cm112-2-6. Google Scholar

[4]

H. Cohen, "A Course in Computational Algebraic Number Theory,'', Graduate Texts in Mathematics, 138 (1993). Google Scholar

[5]

E. C. Dade, O. Taussky and H. Zassenhaus, On the theory of orders, in particular on the semigroup of ideal classes and genera of an order in an algebraic number field,, Math. Ann., 148 (1962), 31. doi: 10.1007/BF01438389. Google Scholar

[6]

F. Grunewald and D. Segal, Some general algorithms. I: Arithmetic groups,, Ann. Math. (2), 112 (1980), 531. Google Scholar

[7]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'', With a supplementary chapter by Katok and Leonardo Mendoza, 54 (1995). Google Scholar

[8]

A. Katok, S. Katok and K. Schmidt, Rigidity of measurable structures for $\mathbbZ^d$-actions by automorphims of a torus,, Comment. Math. Helv., 77 (2002), 718. doi: 10.1007/PL00012439. Google Scholar

[9]

G. Kolutsky, Geometric continued fractions as invariants in the topological classification of Anosov diffeomorphisms of tori,, in, 354 (2009), 99. Google Scholar

[10]

D. Lind and K. Schmidt, Homoclinic points of algebraic $\mathbb Z^d$-actions,, J. Amer. Math. Soc., 12 (1999), 953. doi: 10.1090/S0894-0347-99-00306-9. Google Scholar

[11]

P. Martins Rodrigues and J. Sousa Ramos, Algebraic results on Markov shifts of torus automorphisms,, in, (1998), 436. Google Scholar

[12]

P. Martins Rodrigues and J. Sousa Ramos, Symbolic representations and $\mathbbT^n$ automorphisms,, Int. J. of Bifurcation and Chaos Appl. Sci. Engrg., 13 (2003), 2005. doi: 10.1142/S0218127403007849. Google Scholar

[13]

P. Martins Rodrigues and J. Sousa Ramos, Bowen-Franks groups as conjugacy invariants for $\mathbbT^n$-automorphisms,, Aequationes Math., 69 (2005), 231. doi: 10.1007/s00010-004-2753-7. Google Scholar

[14]

M. Newman, "Integral Matrices,'', Pure and Applied Mathematics, 45 (1972). Google Scholar

[15]

L. Ribes and P. Zalesskii, "Profinite Groups,'', Ergebnisse der Mathematik und ihrer Grenzgebiete, 40 (2000). Google Scholar

[16]

H. P. F. Swinnerton-Dyer, "A Brief Guide to Algebraic Number Theory,'', London Mathematical Society Student Texts, 50 (2001). Google Scholar

[17]

O. Taussky, Connections between algebraic number theory and integral matrices,, an appendix in H. Cohn, (1978). Google Scholar

[18]

D. I. Wallace, Conjugacy classes of hyperbolic matrices in $SL(n,\mathbbZ)$ and ideal classes in an order,, Trans. Amer. Math. Soc., 283 (1984), 177. Google Scholar

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