The corank of a flow over the category of linearly compact vector spaces

TL;DR

This paper explores the structure of topological flows over linearly compact vector spaces, linking entropy to Bernoulli shifts and introducing a corank measure that aligns with topological entropy, providing new insights and proofs.

Contribution

It introduces a notion of corank for topological flows that matches the topological entropy and offers an alternative proof of the Bridge Theorem using Lefschetz duality.

Findings

Topological flows decompose into Bernoulli shifts based on entropy.

Corank effectively measures the complexity of flows in linearly compact spaces.

An alternative proof of the Bridge Theorem connects topological and algebraic entropy.

Abstract

For a topological flow $(V,\phi)$ - i.e., $V$ is a linearly compact vector space and $\phi$ a continuous endomorphism of $V$ - we gain a deep understanding of the relationship between $(V,\phi)$ and the Bernoulli shift: a topological flow $(V,\phi)$ is essentially a product of one-dimensional left Bernoulli shifts as many as $\mathrm{ent}^*(V,\phi)$ counts. This novel comprehension brings us to introduce a notion of corank for topological flows designed for coinciding with the value of the topological entropy of $(V,\phi)$. As an application, we provide an alternative proof of the so-called Bridge Theorem for locally linearly compact vector spaces connecting the topological entropy to the algebraic entropy by means of Lefschetz duality.