Affine transformation with zero entropy and nilsystems

TL;DR

This paper characterizes affine transformations with zero entropy on tori, nilmanifolds, and compact abelian groups, showing they are equivalent to inverse limits of nilsystems based on orbit closure structures.

Contribution

It establishes a precise equivalence between zero entropy affine systems and inverse limits of nilsystems via orbit closure analysis.

Findings

Affine systems with zero entropy have orbit closures isomorphic to inverse limits of nilsystems.

Zero entropy affine transformations are characterized by their orbit closure structures.

The paper provides a structural criterion for zero entropy in affine transformations on specific spaces.

Abstract

In this paper, we study affine transformations on tori, nilmanifolds and compact abelian groups. For these systems, we show that an equivalent condition for zero entropy is the orbit closure of each point has a nice structure. To be precise, the affine systems on those spaces are zero entropy if and only if the orbit closure of each point is isomorphic to an inverse limit of nilsystems.