The Pinsker subgroup of an algebraic flow

TL;DR

This paper investigates the growth behavior of algebraic entropy in abelian group endomorphisms, characterizing the Pinsker subgroup as the maximal invariant subgroup with zero entropy and connecting it to ergodic theory.

Contribution

It introduces a new characterization of the Pinsker subgroup via quasi-periodic points and links algebraic entropy with ergodic properties in compact abelian groups.

Findings

Growth of trajectories is either polynomial or exponential.

The Pinsker subgroup is the largest invariant subgroup with zero entropy.

Existence of a maximal invariant subgroup where the endomorphism is ergodic.

Abstract

The algebraic entropy h, defined for endomorphisms f of abelian groups G, measures the growth of the trajectories of non-empty finite subsets F of G with respect to f. We show that this growth can be either polynomial or exponential. The greatest f-invariant subgroup of G where this growth is polynomial coincides with the greatest f-invariant subgroup P(G,f) of G (named Pinsker subgroup of f) such that h(f|_P(G,f))=0. We obtain also an alternative characterization of P(G,f) from the point of view of the quasi-periodic points of f. This gives the following application in ergodic theory: for every continuous injective endomorphism g of a compact abelian group K there exists a largest g-invariant closed subgroup N of K such that g|_N is ergodic; furthermore, the induced endomorphism g' of the quotient K/N has zero topological entropy.