In dynamical systems theory, the baker's map is a chaotic map from the unit square into itself. It is named after a kneading operation that bakers apply to dough: the dough is cut in half, and the two halves are stacked on one another, and compressed.
There are two alternative definitions of the baker's map which are in common use. One definition folds over or rotates one of the sliced halves before joining it (similar to the horseshoe map) and the other does not.
The folded baker's map acts on the unit square as
When the upper section is not folded over, the map may be written as
The folded baker's map is a two-dimensional analog of the tent map
while the unfolded map is analogous to the Bernoulli map. Both maps are topologically conjugate. The Bernoulli map can be understood as the map that progressively lops digits off the dyadic expansion of x. Unlike the tent map, the baker's map is invertible.
The transfer operator maps functions on the unit square to other functions on the unit square; it is given by
The transfer operator is unitary on the Hilbert space of square-integrable functions on the unit square. The spectrum is continuous, and because the operator is unitary the eigenvalues lie on the unit circle. The transfer operator is not unitary on the space of functions polynomial in the first coordinate and square-integrable in the second. On this space, it has a discrete, non-unitary, decaying spectrum.
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
where each position in the string may take one of the two binary values . The action of the shift operator on this string is
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 is seen to be the unfolded baker's map given above.
Ronald J. Fox, "Construction of the Jordan basis for the Baker map", Chaos, 7 p 254 (1997) doi:10.1063/1.166226
Dean J. Driebe, Fully Chaotic Maps and Broken Time Symmetry, (1999) Kluwer Academic Publishers, Dordrecht Netherlands ISBN0-7923-5564-4(Exposition of the eigenfunctions the Baker's map).