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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, 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), 51-68.  Google Scholar

[2]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Americ. Math Soc., 153 (1971), 401-414. doi: 10.1090/S0002-9947-1971-0274707-X.  Google Scholar

[3]

T. Ferraiol, "Entropia e Ações de Grupos de Lie," Master thesis, University of Campinas, 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), 923-947. 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), 13-27. doi: 10.1007/BF02773793.  Google Scholar

[6]

M. Handel and B. Kitchens, Metrics and entropy for non-compact spaces, Isr. J. Math., 91 (1995), 253-271. doi: 10.1007/BF02761650.  Google Scholar

[7]

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

[8]

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

[9]

M. Patrão, Entropy and its Variational Principle for Non-Compact Metric Spaces, Ergodic Theory and Dynamical Systems, 30 (2010), 1529-1542. 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), 123-136. 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), 768-771.  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), 51-68.  Google Scholar

[2]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Americ. Math Soc., 153 (1971), 401-414. doi: 10.1090/S0002-9947-1971-0274707-X.  Google Scholar

[3]

T. Ferraiol, "Entropia e Ações de Grupos de Lie," Master thesis, University of Campinas, 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), 923-947. 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), 13-27. doi: 10.1007/BF02773793.  Google Scholar

[6]

M. Handel and B. Kitchens, Metrics and entropy for non-compact spaces, Isr. J. Math., 91 (1995), 253-271. doi: 10.1007/BF02761650.  Google Scholar

[7]

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

[8]

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

[9]

M. Patrão, Entropy and its Variational Principle for Non-Compact Metric Spaces, Ergodic Theory and Dynamical Systems, 30 (2010), 1529-1542. 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), 123-136. 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), 768-771.  Google Scholar

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