The Furstenburg Conjecture and Rigidity

Dynamics studies the iterates of a single transformation, or more generally the joint action of several transformations. One of the most striking differences between these is a range of phenomena loosely grouped under the rubric of rigidity.

A single transformation typically has no rigidity, in the sense that it posessed a multitude of invariant measures, or a continuum of compact invariant sets, and so on. However, it often happens that under mild hypotheses the joint action of several transformations will have very few such invariant objects, often only one, but at least they have a simple classification.

For example, a single automorphism of a finite-dimensional torus has infinitely many non-atomic invariant measures, and infinitely many uncountable compact invariant sets. At the other extreme, for action of the group $GL(n,\mathbb{Z})$ on the $n$-torus, the only compact invariant sets are either finite or all of the torus, and the only invariant measures are Lebesgue measure and linear combinations of measures supported on finite sets.

One of the most active areas in dynamics today is finding the boundaries of rigidity behavior, and Furstenberg's Conjecture lies in the intermediate area where rigidity might possibly continue to hold, but no one is sure. Roughly speaking, it says that if two commuting endomorphisms of a torus are incommensurable (no power of one is a power of the other), then the their joint action should be rigid, in the sense that the compact invariant sets and invariant measures should be the obvious algebraic ones. Furstenberg himself [ MR 35 #4369 ] showed that on the one-dimensional circle the only compact sets invariant under both multiplication by $p$ and by $q$, where $\log p$ and $\log q$ are irrationally related, are the whole circle and finite sets (whose elements are necessarily roots of unity). The problem whether Lebesgue measure is the only atomless probability measure invariant under these transformations has come to be known as Furstenberg's Conjecture.

This conjecture is a special case of more general rigidity conjectures in the dynamics of commuting group endomorphisms.

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