Topological entropy for locally linearly compact vector spaces

TL;DR

This paper introduces a new concept of topological entropy for continuous endomorphisms of locally linearly compact vector spaces, establishing fundamental properties and connecting it to algebraic entropy via duality.

Contribution

It defines topological entropy in this new setting, proves the Addition Theorem, and links it to algebraic entropy through a Bridge Theorem, extending entropy theory.

Findings

Topological entropy is additive over short exact sequences.

Established a Bridge Theorem connecting topological and algebraic entropy.

Proved fundamental properties of the new entropy concept.

Abstract

In analogy to the topological entropy for continuous endomorphisms of totally disconnected locally compact groups, we introduce a notion of topological entropy for continuous endomorphisms of locally linearly compact vector spaces. We study the fundamental properties of this entropy and we prove the Addition Theorem, showing that the topological entropy is additive with respect to short exact sequences. By means of Lefschetz Duality, we connect the topological entropy to the algebraic entropy in a Bridge Theorem.