Dyadic transformation

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xy plot where is rational and y = xn for all n.
xy plot where x=x_0 \in [0, 1) is rational and y = xn for all n.

The dyadic transformation (also known as the dyadic map, 2x mod 1 map or Bernoulli map) is the mapping d: \R \to \R^\infty, produced by the rule x_0 \in [0, 1) and x_{n+1} = 2x_n \mod 1 for all n ≥ 0.

Equivalently, the dyadic transformation can also be defined as the iterated function map of the piecewise linear function

f(x)=\begin{cases}2x & 0 \le x < 0.5 \\2x-1 & 0.5 \le x < 1 \end{cases}

The dyadic transformation provides an example of how a simple 1-dimensional map can give rise to chaos.

If x0 is rational the image of x0 contains a finite number of distinct values within [0, 1) and the forward orbit of x0 is eventually periodic, with period equal to the period of the binary expansion of x0. For example, the forward orbit of 11/24 is:

\frac{11}{24} \to \frac{11}{12} \to \frac{5}{6} \to \frac{2}{3} \to \frac{1}{3} \to \frac{2}{3} \to \frac{1}{3} \to \cdots

If x0 is irrational the image of x0 contains an infinite number of distinct values and the forward orbit of x0 is never periodic.

Within any sub-interval of [0,1), no matter how small, there are therefore an infinite number of points whose orbits are eventually periodic, and an infinite number of points whose orbits are never periodic. This sensitivite dependence on initial conditions is a characteristic of chaotic maps.

The dyadic transformation is topologically conjugate to the unit-height tent map.

[edit] Solvability

The dyadic transformation is an exactly solvable model in the theory of deterministic chaos. The square-integrable eigenfunctions of the associated transfer operator of the Bernoulli map are the Bernoulli polynomials. These eigenfunctions form a discrete spectrum with eigenvalues 2 − n for non-negative integers n. There are more general eigenvectors, which are not square-integrable, associated with a continuous spectrum. These are given by the Hurwitz zeta function; equivalently, linear combinations of the Hurwitz zeta give fractal, differentiable-nowhere eigenfunctions, including the Takagi function. The fractal eigenfunctions show a symmetry under the fractal groupoid of the modular group.

[edit] See also

[edit] References

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