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.
The baker's map can be understood as the bilateral shift operator of a bi-infinite two-state lattice model. The baker's map is topologically conjugate to the horseshoe map. In physics, a chain of coupled baker's maps can be used to model deterministic diffusion.
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 an exactly solvable model 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 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 baker's map preserves the two-dimensional Lebesgue measure.
The map is strong mixing and it is topologically mixing.
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.
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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. In all usual cases, the largest eigenvalue is 1, and the corresponding eigenvector is the invariant measure of the system. The transfer operator is sometimes called the Ruelle operator, after David Ruelle, or the Perron–Frobenius operator or Ruelle–Perron–Frobenius operator, in reference to the applicability of the Perron–Frobenius theorem to the determination of the eigenvalues of the operator.
In symbolic dynamics and related branches of mathematics, a shift space or subshift is a set of infinite words that represent the evolution of a discrete system. In fact, shift spaces and symbolic dynamical systems are often considered synonyms. The most widely studied shift spaces are the subshifts of finite type and the sofic shifts. In the classical framework a shift space is any subset of , where is a finite set, which is closed for the Tychonov topology and invariant by translations.
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 (evolution) given by the shift operator. Formally, a Markov partition is used to provide a finite cover for the smooth system; each set of the cover is associated with a single symbol, and the sequences of symbols result as a trajectory of the system moves from one covering set to another.
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