Dots and Laurent Series

At the IHP, Jayadev Athreya spoke about his work with A. Abrams, E. Harris and G. Whitney. This was part of this event.

Shift space

Let \(u \in \{0,1\}^{\mathbb{Z}}\). We think of \(u = \dots u_{-1} u_{0} u_{1} \dots\). We define \((\mathfrak{s}(u))_{i} = u_{i+1}\) to be the shift map.

We think of \(\{0,1\} = \mathbb{F}_{2}\) the field with two elements. We associate a Laurent series for (some?) \(u\) as \[\begin{equation} u \mapsto \sum_{i \in \mathbb{Z}}^{} u_{i} t^{i} \end{equation}\]

We can then consider the more complicated setup with \(\{0,1\}^{\mathbb{Z}^{2}}\). For a sequence \((w_{ij}) \in \{0,1\}^{\mathbb{Z}^{2}}\), one has two shift maps \(\mathfrak{s},\mathfrak{t} \curvearrowright\{0,1\}^{\mathbb{Z}^{2}}\). We can also associate Laurent series \[\begin{equation} w \mapsto \sum_{}^{} w_{ij} s^{i} t^{j} \end{equation}\]

Ledrappier introduced the following set \[\begin{equation} \{w \in \{0,1\}^{\mathbb{Z}^{2}} \mid w_{i,j} + w_{i+1,j} + w_{i,j+1} \equiv 0\pmod{2} \}. \end{equation}\]

This is interesting because it has something to do with mixing (which I don’t understand). Jayadev drew an example below.

Because of the property \[\begin{equation} w_{i,j} + w_{i+1,j} + w_{i,j+1} \equiv 0\pmod{2} \end{equation}\] one has that \[\begin{equation} (1 + s^{-1} + t^{-1}) \sum_{}^{} w_{i,j} s^{i} t^{j} = \sum_{}^{} w_{i,j} s^{i} t^{j} \end{equation}\]

Then we can study the kernel of this multiplication map.

After that, the talk went briefly into the direction of what happens in the case of periodic sequences.