Algebraic and topological entropy on lie groups

Abstract

This paper starts with some examples and quick results on the topological entropy of continuous functions. It discusses the topological entropy on Lie groups and proves their shift properties. It proves Fried's conjecture h( φγ) <- h( φ)+ h( γ) for affine maps on Lie groups. Moreover, φ and γ do not have to commute. As a corollary, it proves that entropy is invariant with isometric endomorphisms of Lie groups. Also, it discusses algebraic entropy on elementary Abelian groups and Lie groups. It proves that the topological entropy is preserved when projected from Lie group lib to its quotient space compact Lie group S 1 for continuous functions lifted from the quotient space and shows that algebraic entropy in general is strictly less than topological entropy.