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The topological entropy of endomorphisms of Lie groups

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Abstract

In this paper, we determine the topological entropy h(φ) of a continuous endomorphism φ of a Lie group G. This computation is a classical topic in ergodic theory which seemed to have long been solved. But, when G is noncompact, the well known Bowen’s formula for the entropy hd(φ) associated to a left invariant distance d just provides an upper bound to h(φ), which is characterized by the so called variational principle. We prove that

$$h(\phi ) = h(\phi {|_{T({G_\phi })}})$$

where Gφ is the maximal connected subgroup of G such that φ(Gφ) = Gφ, and T(Gφ) is the maximal torus in the center of Gφ. This result shows that the computation of the topological entropy of a continuous endomorphism of a Lie group reduces to the classical formula for the topological entropy of a continuous endomorphism of a torus. Our approach explores the relation between null topological entropy and the nonexistence of Li-Yorke pairs and also relies strongly on the structure theory of Lie groups.

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Authors and Affiliations

  1. Department of Mathematics, University of Brasília, Campus Darcy Ribeiro, CEP 70910-900, Brasília, DF, Brazil

    Mauro Patraõ

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  1. Mauro Patraõ
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Correspondence to Mauro Patraõ.

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Patraõ, M. The topological entropy of endomorphisms of Lie groups. Isr. J. Math. 234, 55–80 (2019). https://doi.org/10.1007/s11856-019-1910-6

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  • DOI: https://doi.org/10.1007/s11856-019-1910-6

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