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April  2012, 32(4): 1209-1229. doi: 10.3934/dcds.2012.32.1209

Symbolic approach and induction in the Heisenberg group

1. 

Institut de Mathematiques de Luminy (UMR 6206), Université de la Méditerranee, Campus de Luminy, 13288 MARSEILLE Cedex 9, France

Received  January 2010 Revised  August 2011 Published  October 2011

We associate a homomorphism in the Heisenberg group to each hyperbolic unimodular automorphism of the free group on two generators. We show that the first return-time of some flows in "good" sections, are conjugate to niltranslations, which have the property of being self-induced.
Citation: Jean-Francois Bertazzon. Symbolic approach and induction in the Heisenberg group. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1209-1229. doi: 10.3934/dcds.2012.32.1209
References:
[1]

R. L. Adler, Symbolic dynamics and Markov partitions, Bull. Amer. Math. Soc. (N.S.), 35 (1998), 1-56.

[2]

L. Ambrosio and S. Rigot, Optimal mass transportation in the Heisenberg group, J. Funct. Anal., 208 (2004), 261-301. doi: 10.1016/S0022-1236(03)00019-3.

[3]

P. Arnoux, J. Bernat and X. Bressaud, "Geometric Models for Substitution," Experimental Mathematics, 2010.

[4]

P. Arnoux and C. Mauduit, Complexité de suites engendrées par des récurrences unipotentes, Acta Arithmetica, 76 (1996), 85-97.

[5]

P. Arnoux and A. Siegel, Dynamique du nombre d'or, To appear in Actes de l'Université d'été de Bordeaux, 2004.

[6]

L. Auslander, L. Green and F. Hahn, "Flows on Homogeneous Spaces," With the assistance of L. Markus and W. Massey, and an appendix by L. Greenberg, Annals of Mathematics Studies, 53, Princeton University Press, Princeton, NJ, 1963.

[7]

N. Chekhova, P. Hubert and A. Messaoudi, Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci, J. Théor. Nombres Bordeaux, 13 (2001), 371-394. doi: 10.5802/jtnb.328.

[8]

L. Flaminio and G. Forni, Equidistribution of nilflows and applications to theta sums, Ergodic Theory Dynam. Systems, 26 (2006), 409-433. doi: 10.1017/S014338570500060X.

[9]

P. Fogg, "Substitutions in Dynamics, Arithmetics and Combinatorics," Lecture Notes in Mathematics, 1794, Springer-Verlag, Berlin, 2002.

[10]

H. Furstenberg, Strict ergodicity and transformation of the torus, Amer. J. Math., 83 (1961), 573-601. doi: 10.2307/2372899.

[11]

G. Gelbrich, Self-similar periodic tilings on the Heisenberg group, J. Lie Theory, 4 (1994), 31-37.

[12]

M. Goze and P. Piu, Classification des métriques invariantes à gauche sur le groupe de Heisenberg, Rend. Circ. Mat. Palermo (2), 39 (1990), 299-306. doi: 10.1007/BF02844764.

[13]

L. W. Green, Spectra of nilflows, Bull. Amer. Math. Soc., 67 (1961), 414-415. doi: 10.1090/S0002-9904-1961-10650-2.

[14]

, J. R. Lee and A. Naor, $L_p$ metrics on the Heisenberg group and the Goemans-Linial conjecture.

[15]

E. Lesigne, Sur une nil-variété, les parties minimales associées à une translation sont uniquement ergodiques, Ergodic Theory Dynam. Systems, 11 (1991), 379-391.

[16]

P. Pansu, Plongements quasiisométriques du groupe de Heisenberg dans $L^p$, d'après Cheeger, Kleiner, Lee, Naor, in "Actes du Séminarie de Théorie Spectrale et Géométrie," Vol. 25, Année 2006-2007, 159-176, Sémin. Théor. Spectr. Géom., 25, Univ. Grenoble I, Saint-Martin-d'Hères, 2008.

[17]

M. Queffélec, "Substitution Dynamical Systems-Spectral Analysis," Lecture Notes in Mathematics, 1294, Springer-Verlag, Berlin, 1987.

show all references

References:
[1]

R. L. Adler, Symbolic dynamics and Markov partitions, Bull. Amer. Math. Soc. (N.S.), 35 (1998), 1-56.

[2]

L. Ambrosio and S. Rigot, Optimal mass transportation in the Heisenberg group, J. Funct. Anal., 208 (2004), 261-301. doi: 10.1016/S0022-1236(03)00019-3.

[3]

P. Arnoux, J. Bernat and X. Bressaud, "Geometric Models for Substitution," Experimental Mathematics, 2010.

[4]

P. Arnoux and C. Mauduit, Complexité de suites engendrées par des récurrences unipotentes, Acta Arithmetica, 76 (1996), 85-97.

[5]

P. Arnoux and A. Siegel, Dynamique du nombre d'or, To appear in Actes de l'Université d'été de Bordeaux, 2004.

[6]

L. Auslander, L. Green and F. Hahn, "Flows on Homogeneous Spaces," With the assistance of L. Markus and W. Massey, and an appendix by L. Greenberg, Annals of Mathematics Studies, 53, Princeton University Press, Princeton, NJ, 1963.

[7]

N. Chekhova, P. Hubert and A. Messaoudi, Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci, J. Théor. Nombres Bordeaux, 13 (2001), 371-394. doi: 10.5802/jtnb.328.

[8]

L. Flaminio and G. Forni, Equidistribution of nilflows and applications to theta sums, Ergodic Theory Dynam. Systems, 26 (2006), 409-433. doi: 10.1017/S014338570500060X.

[9]

P. Fogg, "Substitutions in Dynamics, Arithmetics and Combinatorics," Lecture Notes in Mathematics, 1794, Springer-Verlag, Berlin, 2002.

[10]

H. Furstenberg, Strict ergodicity and transformation of the torus, Amer. J. Math., 83 (1961), 573-601. doi: 10.2307/2372899.

[11]

G. Gelbrich, Self-similar periodic tilings on the Heisenberg group, J. Lie Theory, 4 (1994), 31-37.

[12]

M. Goze and P. Piu, Classification des métriques invariantes à gauche sur le groupe de Heisenberg, Rend. Circ. Mat. Palermo (2), 39 (1990), 299-306. doi: 10.1007/BF02844764.

[13]

L. W. Green, Spectra of nilflows, Bull. Amer. Math. Soc., 67 (1961), 414-415. doi: 10.1090/S0002-9904-1961-10650-2.

[14]

, J. R. Lee and A. Naor, $L_p$ metrics on the Heisenberg group and the Goemans-Linial conjecture.

[15]

E. Lesigne, Sur une nil-variété, les parties minimales associées à une translation sont uniquement ergodiques, Ergodic Theory Dynam. Systems, 11 (1991), 379-391.

[16]

P. Pansu, Plongements quasiisométriques du groupe de Heisenberg dans $L^p$, d'après Cheeger, Kleiner, Lee, Naor, in "Actes du Séminarie de Théorie Spectrale et Géométrie," Vol. 25, Année 2006-2007, 159-176, Sémin. Théor. Spectr. Géom., 25, Univ. Grenoble I, Saint-Martin-d'Hères, 2008.

[17]

M. Queffélec, "Substitution Dynamical Systems-Spectral Analysis," Lecture Notes in Mathematics, 1294, Springer-Verlag, Berlin, 1987.

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