American Institute of Mathematical Sciences

Advanced
April  2013, 33(4): 1351-1363. doi: 10.3934/dcds.2013.33.1351

Entropy of endomorphisms of Lie groups

André Caldas 1, and  Mauro Patrão 1,

1. 

Departamento de Matemática, Universidade de Brasília, Campus Darcy Ribeiro, Cx. Postal 4481, Brasília-DF, 70.904-970, Brazil

Received  June 2011 Revised  August 2012 Published  October 2012

We show, when $G$ is a nilpotent or reductive Lie group, that the entropy of any surjective endomorphism coincides with the entropy of its restriction to the toral component of the center of $G$. In particular, if $G$ is a semi-simple Lie group, the entropy of any surjective endomorphism always vanishes. Since every compact group is reductive, the formula for the entropy of a endomorphism of a compact group reduces to the formula for the entropy of an endomorphism of a torus. We also characterize the recurrent set of conjugations of linear semi-simple Lie groups.
Keywords: Topological entropy, endomorphisms of Lie groups, Li-Yorke pairs..
Mathematics Subject Classification: Primary: 37B40, 22D40; Secondary: 37A35, 22E9.
Citation: André Caldas, Mauro Patrão. Entropy of endomorphisms of Lie groups. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1351-1363. doi: 10.3934/dcds.2013.33.1351
References:
[1]

F. Blanchard, E. Glasner, S. Kolyada and A. Maas, On Li-Yorke pairs,, J. Reine Angew. Math., 547 (2002), 51.   Google Scholar

[2]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces,, Trans. Americ. Math Soc., 153 (1971), 401.  doi: 10.1090/S0002-9947-1971-0274707-X.  Google Scholar

[3]

T. Ferraiol, "Entropia e Ações de Grupos de Lie,", Master thesis, (2008).   Google Scholar

[4]

T. Ferraiol, M. Patrão and L. Seco, Jordan decomposition and dynamics on flag manifolds,, Discrete Contin. Dyn. Syst. A, 26 (2010), 923.  doi: 10.3934/dcds.2010.26.923.  Google Scholar

[5]

E. Glasner, A simple characterization of the set of $\mu$-entropy pairs and applications,, Isr. J. Math., 102 (1997), 13.  doi: 10.1007/BF02773793.  Google Scholar

[6]

M. Handel and B. Kitchens, Metrics and entropy for non-compact spaces,, Isr. J. Math., 91 (1995), 253.  doi: 10.1007/BF02761650.  Google Scholar

[7]

S. Helgason, "Differential Geometry, Lie Groups and Symmetric Spaces,", Academic Press, (1978).   Google Scholar

[8]

A. W. Knapp, "Lie Groups Beyond an Introduction,", Progress in Mathematics, 140 (2002).   Google Scholar

[9]

M. Patrão, Entropy and its Variational Principle for Non-Compact Metric Spaces,, Ergodic Theory and Dynamical Systems, 30 (2010), 1529.  doi: 10.1017/S0143385709000674.  Google Scholar

[10]

M. Patrão, L. Santos and L. Seco, A Note on the Jordan Decomposition,, Proyecciones Journal of Mathematics, 30 (2011), 123.  doi: 10.4067/S0716-09172011000100011.  Google Scholar

[11]

Ya. G. Sinai, On the Notion of Entropy of a Dynamical System,, Doklady of Russian Academy of Sciences, 124 (1959), 768.   Google Scholar

show all references

References:
[1]

F. Blanchard, E. Glasner, S. Kolyada and A. Maas, On Li-Yorke pairs,, J. Reine Angew. Math., 547 (2002), 51.   Google Scholar

[2]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces,, Trans. Americ. Math Soc., 153 (1971), 401.  doi: 10.1090/S0002-9947-1971-0274707-X.  Google Scholar

[3]

T. Ferraiol, "Entropia e Ações de Grupos de Lie,", Master thesis, (2008).   Google Scholar

[4]

T. Ferraiol, M. Patrão and L. Seco, Jordan decomposition and dynamics on flag manifolds,, Discrete Contin. Dyn. Syst. A, 26 (2010), 923.  doi: 10.3934/dcds.2010.26.923.  Google Scholar

[5]

E. Glasner, A simple characterization of the set of $\mu$-entropy pairs and applications,, Isr. J. Math., 102 (1997), 13.  doi: 10.1007/BF02773793.  Google Scholar

[6]

M. Handel and B. Kitchens, Metrics and entropy for non-compact spaces,, Isr. J. Math., 91 (1995), 253.  doi: 10.1007/BF02761650.  Google Scholar

[7]

S. Helgason, "Differential Geometry, Lie Groups and Symmetric Spaces,", Academic Press, (1978).   Google Scholar

[8]

A. W. Knapp, "Lie Groups Beyond an Introduction,", Progress in Mathematics, 140 (2002).   Google Scholar

[9]

M. Patrão, Entropy and its Variational Principle for Non-Compact Metric Spaces,, Ergodic Theory and Dynamical Systems, 30 (2010), 1529.  doi: 10.1017/S0143385709000674.  Google Scholar

[10]

M. Patrão, L. Santos and L. Seco, A Note on the Jordan Decomposition,, Proyecciones Journal of Mathematics, 30 (2011), 123.  doi: 10.4067/S0716-09172011000100011.  Google Scholar

[11]

Ya. G. Sinai, On the Notion of Entropy of a Dynamical System,, Doklady of Russian Academy of Sciences, 124 (1959), 768.   Google Scholar

[1]

Ghassen Askri. Li-Yorke chaos for dendrite maps with zero topological entropy and ω-limit sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 2957-2976. doi: 10.3934/dcds.2017127
[2]

Jakub Šotola. Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5119-5128. doi: 10.3934/dcds.2018225
[3]

Adriano Da Silva, Alexandre J. Santana, Simão N. Stelmastchuk. Topological conjugacy of linear systems on Lie groups. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3411-3421. doi: 10.3934/dcds.2017144
[4]

Paul Skerritt, Cornelia Vizman. Dual pairs for matrix groups. Journal of Geometric Mechanics, 2019, 11 (2) : 255-275. doi: 10.3934/jgm.2019014
[5]

Gerard Thompson. Invariant metrics on Lie groups. Journal of Geometric Mechanics, 2015, 7 (4) : 517-526. doi: 10.3934/jgm.2015.7.517
[6]

Min Qian, Jian-Sheng Xie. Entropy formula for endomorphisms: Relations between entropy, exponents and dimension. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 367-392. doi: 10.3934/dcds.2008.21.367
[7]

Katrin Gelfert. Lower bounds for the topological entropy. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555
[8]

Jaume Llibre. Brief survey on the topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3363-3374. doi: 10.3934/dcdsb.2015.20.3363
[9]

Isaac A. García, Jaume Giné, Jaume Llibre. Liénard and Riccati differential equations related via Lie Algebras. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 485-494. doi: 10.3934/dcdsb.2008.10.485
[10]

Benjamin Weiss. Entropy and actions of sofic groups. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3375-3383. doi: 10.3934/dcdsb.2015.20.3375
[11]

M. P. de Oliveira. On 3-graded Lie algebras, Jordan pairs and the canonical kernel function. Electronic Research Announcements, 2003, 9: 142-151.

[12]

Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545
[13]

Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547
[14]

Velimir Jurdjevic. Affine-quadratic problems on Lie groups. Mathematical Control & Related Fields, 2013, 3 (3) : 347-374. doi: 10.3934/mcrf.2013.3.347
[15]

Elena Celledoni, Markus Eslitzbichler, Alexander Schmeding. Shape analysis on Lie groups with applications in computer animation. Journal of Geometric Mechanics, 2016, 8 (3) : 273-304. doi: 10.3934/jgm.2016008
[16]

M. F. Newman and Michael Vaughan-Lee. Some Lie rings associated with Burnside groups. Electronic Research Announcements, 1998, 4: 1-3.

[17]

Firas Hindeleh, Gerard Thompson. Killing's equations for invariant metrics on Lie groups. Journal of Geometric Mechanics, 2011, 3 (3) : 323-335. doi: 10.3934/jgm.2011.3.323
[18]

Gregory S. Chirikjian. Information-theoretic inequalities on unimodular Lie groups. Journal of Geometric Mechanics, 2010, 2 (2) : 119-158. doi: 10.3934/jgm.2010.2.119
[19]

Robert L. Griess Jr., Ching Hung Lam. Groups of Lie type, vertex algebras, and modular moonshine. Electronic Research Announcements, 2014, 21: 167-176. doi: 10.3934/era.2014.21.167
[20]

Nikolaos Karaliolios. Differentiable Rigidity for quasiperiodic cocycles in compact Lie groups. Journal of Modern Dynamics, 2017, 11: 125-142. doi: 10.3934/jmd.2017006

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences