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

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.