- Baker's map
In

dynamical systems theory , the**baker's map**is a chaotic map from the unit square into itself. It is named after akneading operation thatbaker s 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 operator of a bi-infinite two-state lattice model. The baker's map istopologically conjugate to thehorseshoe map . Inphysics , a chain of coupled baker's maps can be used to model deterministicdiffusion . ThePoincaré recurrence time 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 operator of the map. The Baker's map is anexactly solvable model ofdeterministic chaos , in that the eigenfunctions and eigenvalues of the transfer operator can be explicitly determined.**Formal definition**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:$S\_mathrm\{baker-folded\}(x,\; y)\; =\; egin\{cases\}(2x,\; y/2)\; mbox\{for\; \}\; 0\; le\; x\; frac\{1\}\{2\}\; \backslash \; (2-2x,\; 1-y/2)\; mbox\{for\; \}\; frac\{1\}\{2\}\; le\; x\; 1\; end\{cases\}$

When the upper section is not folded over, the map may be written as

:$S\_mathrm\{baker-unrotated\}(x,y)=left(2x-leftlfloor\; 2x\; ight\; floor\; ,,,frac\{y+leftlfloor\; 2x\; ight\; floor\; \}\{2\}\; ight)$

The folded baker's map is a two-dimensional analog of the

tent map :$S\_mathrm\{tent\}(x)\; =egin\{cases\}2x\; mbox\{for\; \}\; 0\; le\; x\; frac\{1\}\{2\}\; \backslash \; 2-2x\; mbox\{for\; \}\; frac\{1\}\{2\}\; le\; x\; 1end\{cases\}$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.**Properties**The Baker map preserves the two-dimensional

Lebesgue measure .The map is

strong mixing and it istopologically mixing .The

transfer operator $U$ maps functions of the unit square to other functions on the unit square; it is given by:$left\; [Uf\; ight]\; (x,y)\; =\; (fcirc\; S^\{-1\})\; (x,y)$

The transfer operator is

unitary on theHilbert space ofsquare-integrable function s 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 $mathcal\{P\}\_xotimes\; L^2\_y$ 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 thesymbolic dynamics of a one-dimensional lattice. Consider, for example, the bi-infinite string:$sigma=left(cdots,sigma\_\{-2\},sigma\_\{-1\},sigma\_\{0\},sigma\_\{1\},sigma\_\{2\},cdots\; ight)$

where each position in the string may take one of the two binary values $sigma\_kin\; \{0,1\}$. The action of the shift operator on this string is

:$au(cdots,sigma\_\{k\},sigma\_\{k+1\},sigma\_\{k+2\},cdots)\; =\; (cdots,sigma\_\{k-1\},sigma\_\{k\},sigma\_\{k+1\},cdots)$

that is, each lattice position is shifted over by one to the left. The bi-infinite string may be represented by two real numbers $0le\; x,yle\; 1$ as

:$x(sigma)=sum\_\{k=0\}^\{infty\}sigma\_\{k\}\; 2^\{-(k+1)\}$

and

:$y(sigma)=sum\_\{k=0\}^\{infty\}sigma\_\{-k-1\}\; 2^\{-(k+1)\}$

In this representation, the shift operator has the representation

:$au(x,y)=left(frac\{x+leftlfloor\; 2y\; ight\; floor\; \}\{2\},,,2y-leftlfloor\; 2y\; ight\; floor\; ight)$

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

**References*** 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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