# American Institute of Mathematical Sciences

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

## Entropy of endomorphisms of Lie groups

 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.
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