Mathematics – Dynamical Systems
Scientific paper
2011-08-25
Mathematics
Dynamical Systems
Changed definition of admissible cover. Otherwise, Proposition 2.8 does not hold
Scientific paper
We extend the variational principle for entropies to a class of continuous maps defined on a countable product of locally compact metric spaces, which we call \emph{product type dynamical systems}. This class includes the shifts and is closed under composition. A major consequence is the variational principle for locally compact dynamical systems without the assumption that the dynamical system is proper. We also apply our results to extend some previous formulas for the topological entropy of continuous endomorphisms of connected Lie groups without the hypothesis that the endomorphism is surjective. In the case of a linear semi-simple Lie group and of a simply-connected nilpotent Lie group, we show that the topological entropy of an endomorphism always vanishes. In particular, the entropy of a linear endomorphism of a finite dimensional vector space always vanishes. In the case of a compact Lie group, we prove that its topological entropy of an endomorphism coincides with the topological entropy of its restriction to the maximal connected and compact subgroup of the center.
Caldas André
Patrão Mauro
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