Symbolic approach and induction in the Heisenberg group

Jean-Francois Bertazzon

doi:10.3934/dcds.2012.32.1209

Article Contents

2012, Volume 32, Issue 4: 1209-1229. Doi: 10.3934/dcds.2012.32.1209

This issue Previous Article Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control Next Article Dense area-preserving homeomorphisms have zero Lyapunov exponents

Symbolic approach and induction in the Heisenberg group

Jean-Francois Bertazzon^1,

1.
Institut de Mathematiques de Luminy (UMR 6206), Université de la Méditerranee, Campus de Luminy, 13288 MARSEILLE Cedex 9

Received: January 2010

Revised: August 2011

Published: April 2012

Abstract / Introduction Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 28D10, 37C55.

Citation:

Related Papers

Cited by

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.