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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 & Continuous Dynamical Systems - A, 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.   Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

L. Auslander, L. Green and F. Hahn, "Flows on Homogeneous Spaces,", With the assistance of L. Markus and W. Massey, 53 (1963).   Google Scholar

[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.  doi: 10.5802/jtnb.328.  Google Scholar

[8]

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

[9]

P. Fogg, "Substitutions in Dynamics, Arithmetics and Combinatorics,", Lecture Notes in Mathematics, 1794 (2002).   Google Scholar

[10]

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

[11]

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

[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.  doi: 10.1007/BF02844764.  Google Scholar

[13]

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

[14]

, J. R. Lee and A. Naor,, \emph{$L_p$ metrics on the Heisenberg group and the Goemans-Linial conjecture}., ().   Google Scholar

[15]

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

[16]

P. Pansu, Plongements quasiisométriques du groupe de Heisenberg dans $L^p$, d'après Cheeger, Kleiner, Lee, Naor,, in, 25 (2008), 2006.   Google Scholar

[17]

M. Queffélec, "Substitution Dynamical Systems-Spectral Analysis,", Lecture Notes in Mathematics, 1294 (1987).   Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

L. Auslander, L. Green and F. Hahn, "Flows on Homogeneous Spaces,", With the assistance of L. Markus and W. Massey, 53 (1963).   Google Scholar

[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.  doi: 10.5802/jtnb.328.  Google Scholar

[8]

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

[9]

P. Fogg, "Substitutions in Dynamics, Arithmetics and Combinatorics,", Lecture Notes in Mathematics, 1794 (2002).   Google Scholar

[10]

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

[11]

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

[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.  doi: 10.1007/BF02844764.  Google Scholar

[13]

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

[14]

, J. R. Lee and A. Naor,, \emph{$L_p$ metrics on the Heisenberg group and the Goemans-Linial conjecture}., ().   Google Scholar

[15]

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

[16]

P. Pansu, Plongements quasiisométriques du groupe de Heisenberg dans $L^p$, d'après Cheeger, Kleiner, Lee, Naor,, in, 25 (2008), 2006.   Google Scholar

[17]

M. Queffélec, "Substitution Dynamical Systems-Spectral Analysis,", Lecture Notes in Mathematics, 1294 (1987).   Google Scholar

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