Analogous problems

One sign of a good problem is that it suggests lots of other analogous problems, and this is certainly the case with Furstenburg's Conjecture. The key feature of the conjecture is the joint action of two (or more) commuting algebraic mappings. We give two examples.

Katok and Spatzier ([ MR 97d:58116] and errata in [ MR 99c:58093]), while trying to understand Rudolph's ideas, developed them in the context to two commuting automorphisms $A$ and $B$ of a finite-dimensional torus, which are of course assumed be independent enough to rule out obvious exceptions. They showed, for example, that any measure that is simultaneously invariant under $A$ and $B$, and that has positive entropy with respect to, say, $A$, must be a multiple of Lebesgue measure on the unstable foliation of $A$ on the torus. This does not show directly that the measure is Lebesgue measure in all cases, but does in some cases with further assumptions.

There are also analogous questions for totally disconnected groups, although here the statements are more intricate because of the existence of many more ``obvious'' invariant sets. A typical example is due originally to Ledrappier [ MR 80b:28030]. Let $X$ be the subgroup of $(\mathbb{Z}/2\mathbb{Z})^{\mathbb{Z}^2}$ defined by the condition that $x_{i,j}+x_{i+1,j}+x_{i,j+1}=0$ for all $i,j\in\mathbb{Z}$. Let $A$ shift an array one place to the left, and $B$ shift it one place down, so that $A$ and $B$ are commuting automorphisms of $X$. What are the invariant measures for the joint actions of $A$ and $B$? Here there are many more compact invariant sets than just finite sets, and each supports at least one invariant measure. Nevertheless, there may well be a simple algebraic classification of the invariant measures, even in this case.

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