Symmetry of entropy in higher rank diagonalizable actions and measure classification

TL;DR

This paper explores the symmetry properties of entropy in higher rank algebraic actions, showing that lack of certain entropy symmetries implies invariance under unipotent groups, with implications for measure classification.

Contribution

It establishes a link between entropy symmetry failure and measure invariance under unipotent groups in higher rank algebraic actions.

Findings

Failure of entropy symmetry implies measure invariance under unipotent groups

Measures invariant under multiparameter algebraic actions exhibit specific entropy properties

Consequences for measure classification in dynamical systems

Abstract

An important consequence of the theory of entropy of Z-actions is that the events measurable with respect to the far future coincide (modulo null sets) with those measurable with respect to the distant past, and that measuring the entropy using the past will give the same value as measuring it using the future. In this paper we show that for measures invariant under multiparameter algebraic actions if the entropy attached to coarse Lyapunov foliations fail to display a stronger symmetry property of a similar type this forces the measure to be invariant under non-trivial unipotent groups. Some consequences of this phenomenon are noted.