Approaches to a proof

Essentially all approaches to Furstenburg's Conjecture and related problems have so far been the same, namely the use of isometric directions for the action, where the action acts as a translation on a certain foliation, and invoking the observation that the only translation-invariant measures are Lebesgue on this foliation. The positive entropy assumption is then used to show that this forces the measure itself to be Lebesgue.

For example, if $m$ and $n$ are integers such that $n/m$ is close to $-(\log 2)/(\log 3)$, then $\phi_2^m\circ\phi_3^n$ is nearly an isometry on the circle, or more accurately on the $6$-adic solenoid since we must invert one of the two maps. This solenoid has a copy of the reals wrapping densely through it, and the conditional measures on nearby pieces of leaves induced by an invariant measure must have the property that under iterates in the isometric direction they are nearly translation-invariant. This is the key idea in Rudolph's proof.

The existence of isometric directions for certain $\mathbb{Z}^d$-actions is a special case of a very general phenomenon called ``subdynamics'', or the study of such actions along subgroups of $\mathbb{Z}^d$, or more generally along subspaces of $\mathbb{R}^d$, introduced by Boyle and Lind [ MR 97d:58115]. Every topological $\mathbb{Z}^d$-action on an infinite compact space has a non-empty set of lower-dimension subspaces, closed in the Grassmann topology, along which the action is nonexpansive. Dynamical properties within a connected component of the complement of this set vary nicely or are constant, while passing from one component to another typically results in abrupt changes, roughly analogous to a phase transition. It is these nonexpansive directions that have been the key to all attempts so far to prove Furstenburg's Conjecture.

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