Approaches to a counterexample

Some have expressed serious doubts whether Furstenburg's Conjecture is even true. Mainly this is based on the observation, mentioned above, that every proof so far runs up against the same positive entropy barrier. Either there is a zero entropy counterexample, or a genuinely new idea is needed for a proof.

One approach to constructing an atomless measure, invariant under $\phi_2$ and $\phi_3$, which is not Lebesgue measure is as follows. We start by observing that there is a Markov partition that simultaneously works for both maps, namely the partition of $\mathbb{T}$ into six equal intervals $[j/6,(j+1)/6)$ for $0\le j\le 5$. Start the construction by assigning weights to each of these, giving six numbers $a_0$ through $a_5$. Invariance under $\phi_2$ gives linear relations between the $a_{j}$, and invariance under $\phi_3$ gives further linear relations. Additionally, the $a_j$ must add up to 1. Together these cut the dimension of possible solutions from six to two.

Each interval is subdivided into six equal subintervals, so let each $a_j$ be divided into weights $a_{j0}$ through $a_{j5}$. Again, invariance under $\phi_2$ gives linear relations between the $a_{jk}$, and invariance under $\phi_3$ gives further linear relations. In addition, the sum of the $a_{jk}$ must equal $a_j$. All these together give a set of equalities and inequalities, which can be solved by linear programming software. The result is that the set of solutions is a convex object in a 10-dimensional subspace 36-dimensional space with 876 vertices. Each represents a potential start for a counterexample.

The idea is to try to continue this process a few more levels, to see what it takes for an assignment of weights to be continued to the next level consistently. Some form of the zero entropy hypothesis on the measure (which is certainly necessary by Rudolph's theorem) should guide the iterative construction from one level to the next. In the end, one would end up with an assignment of weights to all the 6-adic intervals, consistent with defining a jointly invariant measure, and for which at some stage not all intervals are given equal weight. To make this measure atomless, one needs to require further that the maximum measure of the intervals at stage $n$ must tend to $0$ as $n\to\infty$.

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