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Baker's map

 

 
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Baker's map
 
 
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In dynamical systems theoryDynamical systems theory

Dynamical systems theory is an area of applied mathematics used to describe the behavior of complex dynamical systems by emp...
, the baker's map is a chaoticChaos theory

In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain co...
 map from the unit square into itself. It is named after a kneadingFacts About Kneading

Kneading is a process in the making of bread, used to mix together the ingredients and add strength to the bread....
 operation that bakerBaker Summary

A baker is someone who primarily bakes and sells bread....
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 operatorShift operator

In mathematics, and in particular functional analysis, the shift operators are examples of linear operators, important for t...
 of a bi-infinite two-state lattice modelLattice model (physics)

In physics, a lattice model is a physical model that is defined on a lattice, as opposed to the continuum of space or spacet...
. The baker's map is topologically conjugate to the horseshoe mapFacts About Horseshoe map

----In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself....
. In physicsPhysics

Physics , the most fundamental physical science, is concerned with the underlying principles of the natural world....
, a chain of coupled baker's maps can be used to model deterministic diffusionDiffusion

Diffusion, being the spontaneous spreading of matter , heat, or momentum, is one type of transport phenomenon....
. The Poincaré 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 operatorTransfer operator

In mathematics, the transfer operator encodes information about an iterated map and is frequently used to study the behavior...
 of the map. The Baker's map is an exactly solvable model of deterministic 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 mapHorseshoe map

----In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself....
) 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 mapTent map

In mathematics, the tent map is an iterated function, in the shape of a tent, forming a discrete-time dynamical system....


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 measureLebesgue measure

In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space...
.

The map is strong mixing and it is topologically mixing.

The transfer operatorTransfer operator

In mathematics, the transfer operator encodes information about an iterated map and is frequently used to study the behavior...
  maps functions of the unit square to other functions on the unit square; it is given by

The transfer operator is unitaryUnitary

Unitary may refer to:* In Christian doctrine, unitarianism is the belief in a "unitary God" as opposed to the concept...
 on the Hilbert spaceHilbert space

In mathematics, a Hilbert space is a generalization of Euclidean space that is not restricted to finite dimensions....
 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, 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 operatorShift operator

In mathematics, and in particular functional analysis, the shift operators are examples of linear operators, important for t...
 on the symbolic dynamicsSymbolic dynamics

In mathematics, symbolic dynamics is the practice of modelling a dynamical system by a space consisting of infinite sequence...
 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 representation

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