• Previous Article
    Dense area-preserving homeomorphisms have zero Lyapunov exponents
  • DCDS Home
  • This Issue
  • Next 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
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

[1]

Heping Liu, Yu Liu. Refinable functions on the Heisenberg group. Communications on Pure & Applied Analysis, 2007, 6 (3) : 775-787. doi: 10.3934/cpaa.2007.6.775

[2]

Isabeau Birindelli, J. Wigniolle. Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2003, 2 (4) : 461-479. doi: 10.3934/cpaa.2003.2.461

[3]

Sze-Bi Hsu, Bernold Fiedler, Hsiu-Hau Lin. Classification of potential flows under renormalization group transformation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 437-446. doi: 10.3934/dcdsb.2016.21.437

[4]

Xinjing Wang, Pengcheng Niu, Xuewei Cui. A Liouville type theorem to an extension problem relating to the Heisenberg group. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2379-2394. doi: 10.3934/cpaa.2018113

[5]

L. Brandolini, M. Rigoli and A. G. Setti. On the existence of positive solutions of Yamabe-type equations on the Heisenberg group. Electronic Research Announcements, 1996, 2: 101-107.

[6]

Pablo Ochoa. Approximation schemes for non-linear second order equations on the Heisenberg group. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1841-1863. doi: 10.3934/cpaa.2015.14.1841

[7]

Luis F. López, Yannick Sire. Rigidity results for nonlocal phase transitions in the Heisenberg group $\mathbb{H}$. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2639-2656. doi: 10.3934/dcds.2014.34.2639

[8]

Patrizia Pucci. Critical Schrödinger-Hardy systems in the Heisenberg group. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 375-400. doi: 10.3934/dcdss.2019025

[9]

Fausto Ferrari, Qing Liu, Juan Manfredi. On the characterization of $p$-harmonic functions on the Heisenberg group by mean value properties. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2779-2793. doi: 10.3934/dcds.2014.34.2779

[10]

I. Moise, Roger Temam. Renormalization group method: Application to Navier-Stokes equation. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 191-210. doi: 10.3934/dcds.2000.6.191

[11]

G. A. Braga, Frederico Furtado, Vincenzo Isaia. Renormalization group calculation of asymptotically self-similar dynamics. Conference Publications, 2005, 2005 (Special) : 131-141. doi: 10.3934/proc.2005.2005.131

[12]

Wenlei Li, Shaoyun Shi. Singular perturbed renormalization group theory and its application to highly oscillatory problems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1819-1833. doi: 10.3934/dcdsb.2018089

[13]

Nathan Glatt-Holtz, Mohammed Ziane. Singular perturbation systems with stochastic forcing and the renormalization group method. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1241-1268. doi: 10.3934/dcds.2010.26.1241

[14]

Chiu-Ya Lan, Chi-Kun Lin. Asymptotic behavior of the compressible viscous potential fluid: Renormalization group approach. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 161-188. doi: 10.3934/dcds.2004.11.161

[15]

Hans Koch. A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 881-909. doi: 10.3934/dcds.2004.11.881

[16]

Houda Mokrani. Semi-linear sub-elliptic equations on the Heisenberg group with a singular potential. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1619-1636. doi: 10.3934/cpaa.2009.8.1619

[17]

Pablo Ochoa, Julio Alejo Ruiz. A study of comparison, existence and regularity of viscosity and weak solutions for quasilinear equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1091-1115. doi: 10.3934/cpaa.2019053

[18]

Vincenzo Michael Isaia. Numerical simulation of universal finite time behavior for parabolic IVP via geometric renormalization group. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3459-3481. doi: 10.3934/dcdsb.2017175

[19]

G. A. Braga, Frederico Furtado, Jussara M. Moreira, Leonardo T. Rolla. Renormalization group analysis of nonlinear diffusion equations with time dependent coefficients: Analytical results. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 699-715. doi: 10.3934/dcdsb.2007.7.699

[20]

Laura Cremaschi, Carlo Mantegazza. Short-time existence of the second order renormalization group flow in dimension three. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5787-5798. doi: 10.3934/dcds.2015.35.5787

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]