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

Abstract

For a topological flow $(V,\varphi )$ - i.e., $V$ is a linearly compact vector space and $\varphi$ a continuous endomorphism of $V$ - we gain a deep understanding of the relationship between $(V,\varphi )$ and the Bernoulli shift: a topological flow $(V,\varphi )$ is essentially a product of one-dimensional left Bernoulli shifts as many as ${\mathrm{ent}}^{\ast }(V,\varphi )$ 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 $(V,\varphi )$. 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.

the organisation of the paper has been substantially changed