Baker's Map - As A Shift Operator

As A Shift Operator

The baker's map can be understood as the two-sided shift operator on the symbolic dynamics of a one-dimensional lattice. Consider, for example, the bi-infinite string

$\sigma=\left(\ldots,\sigma_{-2},\sigma_{-1},\sigma_{0}, \sigma_{1},\sigma_{2},\ldots \right)$

where each position in the string may take one of the two binary values . The action of the shift operator on this string is

$\tau(\ldots,\sigma_{k},\sigma_{k+1},\sigma_{k+2},\ldots) = (\ldots,\sigma_{k-1},\sigma_{k},\sigma_{k+1},\ldots)$

that is, each lattice position is shifted over by one to the left. The bi-infinite string may be represented by two real numbers as

and

In this representation, the shift operator has the form

which can be seen to be the inverse of the un-folded baker's map given above.

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