- Baker's map
dynamical systems theory, the baker's map is a chaotic map from the unit square into itself. It is named after a kneadingoperation that bakers apply to dough: the dough is cut in half, and the two halves are stacked on one-another, and compressed.
The baker's map can be understood as the bilateral
shift operatorof a bi-infinite two-state lattice model. The baker's map is topologically conjugateto the horseshoe map. In physics, a chain of coupled baker's maps can be used to model deterministic diffusion. The Poincaré recurrencetime of the Baker's map is short compared to Hamiltonian maps.
As with many deterministic dynamical systems, the baker's map is studied by its action on the space of functions defined on the unit square. The Baker's map defines an operator on the space of functions, known as the
transfer operatorof the map. The Baker's map is an exactly solvablemodel of deterministic chaos, in that the eigenfunctions and eigenvalues of the transfer operator can be explicitly determined.
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
while the non-rotated 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 Baker map preserves the two-dimensional
The map is
strong mixingand it is topologically mixing.
transfer operatormaps functions of the unit square to other functions on the unit square; it is given by
The transfer operator is
unitaryon the Hilbert spaceof square-integrable functions on the unit square. The spectrum is continuous, and because the operator is unitary, the eigenvalues lie on the unit circle, of course. 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 operatoron the symbolic dynamicsof 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
In this representation, the shift operator has the representation
which can be seen to be the inverse of the un-folded baker's map given above.
* Hiroshi H. Hasagawa and William C. Saphir, "Unitarity and irreversibility in chaotic systems", "Physical Review A", 46, p7401 (1992)
* Ronald J. Fox, "Construction of the Jordan basis for the Baker map", "Chaos", 7 p 254 (1997)
* Dean J. Driebe, "Fully Chaotic Maps and Broken Time Symmetry", (1999) Kluwer Academic Publishers, Dordrecht Netherlands ISBN 0-7923-5564-4 "(Exposition of the eigenfunctions the Baker's map)".
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