In dynamical systems theory

Dynamical systems theory

Dynamical systems theory is an area of applied mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. When difference...

, the baker's map is a chaotic

Chaos theory

Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...

 map from the unit square into itself. It is named after a kneading

Kneading

Kneading is a process in the making of bread or pasta dough, used to mix together the ingredients and add strength to the final product. Its importance lies in the mixing of flour with water. When these two ingredients are combined and kneaded, the gliadin and glutenin proteins in the flour expand...

 operation that baker

Baker

A baker is someone who bakes and sells bread, Cakes and similar foods may also be produced, as the traditional boundaries between what is produced by a baker as opposed to a pastry chef have blurred in recent decades...

s apply to dough: the dough is cut in half, and the two halves are stacked on one-another, and compressed. (The word baker is used to denote the profession and not a name, so it should not be capitalized.)

The baker's map can be understood as the bilateral shift operator

Shift operator

In mathematics, and in particular functional analysis, the shift operator or translation operator is an operator that takes a function to its translation . In time series analysis, the shift operator is called the lag operator....

 of a bi-infinite two-state lattice model

Lattice 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 spacetime. Lattice models originally occurred in the context of condensed matter physics, where the atoms of a crystal automatically form a lattice. Currently, lattice models are...

. The baker's map is topologically conjugate to the horseshoe map

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. It is a core example in the study of dynamical systems. The map was introduced by Stephen Smale while studying the behavior of the orbits of the van der Pol oscillator...

. In physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, a chain of coupled baker's maps can be used to model deterministic diffusion

Diffusion

Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...

. 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 operator

Transfer operator

In mathematics, the transfer operator encodes information about an iterated map and is frequently used to study the behavior of dynamical systems, statistical mechanics, quantum chaos and fractals...

 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 map

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. It is a core example in the study of dynamical systems. The map was introduced by Stephen Smale while studying the behavior of the orbits of the van der Pol oscillator...

) 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 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's map preserves the two-dimensional Lebesgue measure

Lebesgue measure

In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...

.

The map is strong mixing and it is topologically mixing.

The transfer operator

Transfer operator

In mathematics, the transfer operator encodes information about an iterated map and is frequently used to study the behavior of dynamical systems, statistical mechanics, quantum chaos and fractals...

  maps functions of the unit square to other functions on the unit square; it is given by


The transfer operator is unitary

Unitary

Unitary may refer to:* Unitary construction, in automotive design, another common term for a unibody or monocoque construction**Unitary as chemical weapons opposite of Binary...

 on the Hilbert space

Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

 of square-integrable function

Square-integrable function

In mathematics, a quadratically integrable function, also called a square-integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite...

s 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

Shift operator

In mathematics, and in particular functional analysis, the shift operator or translation operator is an operator that takes a function to its translation . In time series analysis, the shift operator is called the lag operator....

 on the symbolic dynamics

Symbolic dynamics

In mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics given by the shift operator...

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