Entropy of endomorphisms of Lie groups

André Caldas; Mauro Patrão

doi:10.3934/dcds.2013.33.1351

Article Contents

2013, Volume 33, Issue 4: 1351-1363. Doi: 10.3934/dcds.2013.33.1351

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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

Received: May 31, 2011

Revised: July 31, 2012

Published: March 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 37B40, 22D40; Secondary: 37A35, 22E99.

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References

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