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April  2021, 41(4): 1543-1559. doi: 10.3934/dcds.2020330

Jordan decomposition and the recurrent set of flows of automorphisms

Víctor Ayala 1,, , Adriano Da Silva 2, and  Philippe Jouan 3,

1. 

Instituto de Alta Investigación, Universidad de Tarapacá, Arica, Chile
2. 

Instituto de Matemática, Universidade Estadual de Campinas, Brazil
3. 

Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, France

* Corresponding author: Víctor Ayala

Received  February 2020 Revised  August 2020 Published  April 2021 Early access  September 2020

Fund Project: Supported by Proyecto Fondecyt n° 1190142. Conicyt, Chile.
Supported by Fapesp grant n° 2018/10696-6

In this paper we show that any linear vector field $ \mathcal{X} $ on a connected Lie group $ G $ admits a Jordan decomposition and the recurrent set of the associated flow of automorphisms is given as the intersection of the fixed points of the hyperbolic and nilpotent components of its Jordan decomposition.

Keywords: Linear vector fields, recurrent points, Jordan decomposition.
Mathematics Subject Classification: 37B20, 54H20, 37B99.
Citation: Víctor Ayala, Adriano Da Silva, Philippe Jouan. Jordan decomposition and the recurrent set of flows of automorphisms. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1543-1559. doi: 10.3934/dcds.2020330
References:
[1]

V. I. Arnold and A. Avez, Ergodic Problems in Classical Mechanics, , New York: Benjamin, 1968.  Google Scholar

[2]

V. Ayala and A. Da Silva, Controllability of linear control systems on Lie groups with semisimple finite center, SIAM Journal on Control and Optimization, 55 (2017), 1332-1343.  doi: 10.1137/15M1053037.  Google Scholar

[3]

V. Ayala and A. Da Silva, On the characterization of the controllability property for linear control systems on nonnilpotent, solvable threedimensional Lie groups, Journal of Differential Equations, 266 (2019), 8233-8257.  doi: 10.1016/j.jde.2018.12.027.  Google Scholar

[4]

V. Ayala, A. Da Silva and G. Zsigmond, Control sets of linear systems on Lie groups, Nonlinear Differential Equations and Applications - NoDEA, 24 (2017), 1-15.  doi: 10.1007/s00030-017-0430-5.  Google Scholar

[5]

V. Ayala, A. Da Silva, P. Jouan and G. Zsigmond, Control sets of linear systems on semi-simple Lie groups, J. Differ. Equ., 269 (2020), 449-466.  doi: 10.1016/j.jde.2019.12.010.  Google Scholar

[6]

V. Ayala and P. Jouan, Almost-riemannian geometry on lie groups, SIAM Journal on Control and Optimization, 54 (2016), 2919-2947.  doi: 10.1137/15M1038372.  Google Scholar

[7]

V. Ayala and J. Tirao, Linear control systems on lie groups and controllability, American Mathematical Society, Series: Symposia in Pure Mathematics, 64 (1999), 47-64.  doi: 10.1090/pspum/064/1654529.  Google Scholar

[8]

A. Da Silva, Controllability of linear systems on solvable Lie groups, SIAM Journal on Control and Optimization, 54 (2016), 372-390.  doi: 10.1137/140998342.  Google Scholar

[9]

T. Ferraiol, M. Patrão and L. Seco, Jordan decomposition and dynamics on flag manifolds, Discrete and Continuous Dynamical Systems - Series A, 26 (2010), 923-947.  doi: 10.3934/dcds.2010.26.923.  Google Scholar

[10]

P. Jouan, Equivalence of control systems with linear systems on lie groups and homogeneous spaces, ESAIM: Control Optimization and Calculus of Variations, 16 (2010), 956-973.  doi: 10.1051/cocv/2009027.  Google Scholar

[11]

V. Kivioja and E. Le Donne, Isometries of nilpotent metric groups, J. École Polytechnique, Mathématiques, Tome 4 (2017), 473–482. doi: 10.5802/jep.48.  Google Scholar

[12]

A. W. Knapp, Lie Groups Beyond an Introduction, Second Edition, Birkhäuser Boston, Inc., Boston, MA, 2002.  Google Scholar

[13]

J. Milnor, Curvatures of left invariant metrics on Lie groups, Advances in Math., 21 (1976), 293-329.  doi: 10.1016/S0001-8708(76)80002-3.  Google Scholar

[14]

G. D. Mostow, Fully reducible subgroups of algebraic groups, American Journal of Mathematics, 78 (1956), 200-221.  doi: 10.2307/2372490.  Google Scholar

[15]

E. Noether, Invariant variation problems, Transp. Theory Statist. Phys., 1 (1971), 186-207.  doi: 10.1080/00411457108231446.  Google Scholar

[16]

M. Patrão, Entropy and its variational principle for non-compact metric spaces, Ergod. Th. & Dynam. Sys., 30 (2010), 1529-1542.  doi: 10.1017/S0143385709000674.  Google Scholar

[17]

M. Patrão, The topological entropy of endomorphisms of Lie groups, Israel Journal of Mathematics, 234 (2019), 55-80.  doi: 10.1007/s11856-019-1910-6.  Google Scholar

[18]

A. L. Onishchik and E. B. Vinberg, Lie Groups and Lie Algebras Ⅲ - Structure of Lie Groups and Lie Algebras, , Berlin: Springer, 1990. doi: 10.1007/978-3-642-74334-4.  Google Scholar

[19]

L. A. B. San Martin, Algebras de Lie, Second Edition, Editora Unicamp, 2010. Google Scholar

show all references

References:
[1]

V. I. Arnold and A. Avez, Ergodic Problems in Classical Mechanics, , New York: Benjamin, 1968.  Google Scholar

[2]

V. Ayala and A. Da Silva, Controllability of linear control systems on Lie groups with semisimple finite center, SIAM Journal on Control and Optimization, 55 (2017), 1332-1343.  doi: 10.1137/15M1053037.  Google Scholar

[3]

V. Ayala and A. Da Silva, On the characterization of the controllability property for linear control systems on nonnilpotent, solvable threedimensional Lie groups, Journal of Differential Equations, 266 (2019), 8233-8257.  doi: 10.1016/j.jde.2018.12.027.  Google Scholar

[4]

V. Ayala, A. Da Silva and G. Zsigmond, Control sets of linear systems on Lie groups, Nonlinear Differential Equations and Applications - NoDEA, 24 (2017), 1-15.  doi: 10.1007/s00030-017-0430-5.  Google Scholar

[5]

V. Ayala, A. Da Silva, P. Jouan and G. Zsigmond, Control sets of linear systems on semi-simple Lie groups, J. Differ. Equ., 269 (2020), 449-466.  doi: 10.1016/j.jde.2019.12.010.  Google Scholar

[6]

V. Ayala and P. Jouan, Almost-riemannian geometry on lie groups, SIAM Journal on Control and Optimization, 54 (2016), 2919-2947.  doi: 10.1137/15M1038372.  Google Scholar

[7]

V. Ayala and J. Tirao, Linear control systems on lie groups and controllability, American Mathematical Society, Series: Symposia in Pure Mathematics, 64 (1999), 47-64.  doi: 10.1090/pspum/064/1654529.  Google Scholar

[8]

A. Da Silva, Controllability of linear systems on solvable Lie groups, SIAM Journal on Control and Optimization, 54 (2016), 372-390.  doi: 10.1137/140998342.  Google Scholar

[9]

T. Ferraiol, M. Patrão and L. Seco, Jordan decomposition and dynamics on flag manifolds, Discrete and Continuous Dynamical Systems - Series A, 26 (2010), 923-947.  doi: 10.3934/dcds.2010.26.923.  Google Scholar

[10]

P. Jouan, Equivalence of control systems with linear systems on lie groups and homogeneous spaces, ESAIM: Control Optimization and Calculus of Variations, 16 (2010), 956-973.  doi: 10.1051/cocv/2009027.  Google Scholar

[11]

V. Kivioja and E. Le Donne, Isometries of nilpotent metric groups, J. École Polytechnique, Mathématiques, Tome 4 (2017), 473–482. doi: 10.5802/jep.48.  Google Scholar

[12]

A. W. Knapp, Lie Groups Beyond an Introduction, Second Edition, Birkhäuser Boston, Inc., Boston, MA, 2002.  Google Scholar

[13]

J. Milnor, Curvatures of left invariant metrics on Lie groups, Advances in Math., 21 (1976), 293-329.  doi: 10.1016/S0001-8708(76)80002-3.  Google Scholar

[14]

G. D. Mostow, Fully reducible subgroups of algebraic groups, American Journal of Mathematics, 78 (1956), 200-221.  doi: 10.2307/2372490.  Google Scholar

[15]

E. Noether, Invariant variation problems, Transp. Theory Statist. Phys., 1 (1971), 186-207.  doi: 10.1080/00411457108231446.  Google Scholar

[16]

M. Patrão, Entropy and its variational principle for non-compact metric spaces, Ergod. Th. & Dynam. Sys., 30 (2010), 1529-1542.  doi: 10.1017/S0143385709000674.  Google Scholar

[17]

M. Patrão, The topological entropy of endomorphisms of Lie groups, Israel Journal of Mathematics, 234 (2019), 55-80.  doi: 10.1007/s11856-019-1910-6.  Google Scholar

[18]

A. L. Onishchik and E. B. Vinberg, Lie Groups and Lie Algebras Ⅲ - Structure of Lie Groups and Lie Algebras, , Berlin: Springer, 1990. doi: 10.1007/978-3-642-74334-4.  Google Scholar

[19]

L. A. B. San Martin, Algebras de Lie, Second Edition, Editora Unicamp, 2010. Google Scholar

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