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I originally posted this on math.stack, but no one answered, so im posting here:
I need help understanding the following construction (Taken from the paper "Entropy and isomorphism theorems for actions of amenable groups"):

Begin by taking the direct product of two dyadic odometers. The space for this $\mathbb{Z^2}$-action can be thought of as the collection of all the different ways of organizing $\mathbb{Z^2}$, first into a tiling by
$2$ x $2$ squares, then a tiling by $4$ x $4$ squares, each well tiled by the $2$ x $2$ squares, then a tiling by $8$ x $8$ squares, etc. $\mathbb{Z^2}$ acts on this space of patterns by translations. Next to each such pattern we will associate to each element of $\mathbb{Z^2}$ a random variable taking the values $+-$ with probability $(1/2,1/2)$. These random variables will be independent, subject to the following constraints:

$(1)$ in every $2$ x $2$ square the bottom pair is equal to the top pair;

$(2)$ in every $4$ x $4$ square, the left pair of $2$ x $2$ squares equals the right pair of $2$ x $2$ squares;

$(3)$ in every $2^3$ x $2^3$ square the bottom pair of $2^2$ x $2^2$ squares equals the top pair of $2^2$ x $2^2$ squares;
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$(2n)$ in every $2^{2n}$ x $2^{2n}$ square the left half of the square equals the the right half of the square;

$(2n+1)$ in every $2^{2n+1}$ x $2^{2n+1}$ square the bottom half of the square equals the top half of the square.
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It is clear that along the $45^∘$ diagonals one sees independent random variables, thus the entropy of $ST$ is at least $\log 2$, while in the vertical and horizontal directions the entropy is killed because we repeat exactly, infinitely many times. Thus $h(S)= h(T)=0$ but $h(ST)≥\log2$.

I don't understand what exactly is the measure space in this construction. Is the underlying set a subset of $\mathrm{odometer}$ x $\mathrm{odometer}$ x $\{+,-\}^{\mathbb{Z^2}}$$?$ What is the measure?
Can someone help me figure it out? it's probably standard.
Thanks.

Here is a picture of the construction, if you prefer.

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In fact, it is not essential to understand the measure. The authors mean the topological entropy of S, T and ST. For ST the entropy is (at least) log 2 because already along diagonal lines you can see all possible 0-1 sequences. In the axial directions, if you consider say horizontal strips of no matter what (vertical) width, after some time you will see repetitions, and this leads (via an easy counting argument) to entropy zero. If you insist on finding a measure which reveals the phenomenon, start by taking the ST-invariant measure \mu of maximal entropy for ST. It has ST-entropy log 2 (by the variational principle). Then, in order to obtain a measure invariant under both S and T, you should average along the orbit of \mu under the double action. The limit measure, say \nu, will maintain entropy log 2 for ST (by upper semi-continuity of entropy in subshifts). While for S and T \nu must have entropy zero, because this entropy is diminated by the topological entropy of S and T. Finally, there exists a relatively easy description of the measure \nu. And it is not odometer x odometer x Bernoulli, because the Z^2-entropy of \nu is in fact zero. It is some skew-product extension of odometer x odometer and can be described by giving exact probabilities of 2^n x 2^n configurations. In fact, that's how the authors implicitely describe the example, by saying that the contents of some squares are "independent". You only need to understand independence stochastically, with equal measures to all admissible cylinders over the same square.

I hope this helps

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