Dyadic transformation
) produced by the rule Equivalently, the dyadic transformation can also be defined as the iterated function map of the piecewise linear function The name bit shift map arises because, if the value of an iterate is written in binary notation, the next iterate is obtained by shifting the binary point one bit to the right, and if the bit to the left of the new binary point is a "one", replacing it with a zero.
The dyadic transformation provides an example of how a simple 1-dimensional map can give rise to chaos.
It was introduced by Alfréd Rényi in 1957, and an invariant measure for it was given by Alexander Gelfond in 1959 and again independently by Bill Parry in 1960.
consisting of all doubly-infinite (double-ended) strings; this will lead to the Baker's map.
Note that the sum gives the Cantor function, as conventionally defined.
One hallmark of chaotic dynamics is the loss of information as simulation occurs.
And since our simulation has reached a fixed point, for almost all initial conditions it will not describe the dynamics in the qualitatively correct way as chaotic.
The dyadic transformation is topologically semi-conjugate to the unit-height tent map.
This is stable under iteration, as It is also conjugate to the chaotic r = 4 case of the logistic map.
That is, the map is given by Because of the simple nature of the dynamics when the iterates are viewed in binary notation, it is easy to categorize the dynamics based on the initial condition: If the initial condition is irrational (as almost all points in the unit interval are), then the dynamics are non-periodic—this follows directly from the definition of an irrational number as one with a non-repeating binary expansion.
That is, the homomorphism is basically a statement that the Cantor set can be mapped into the reals.
It is a surjection: every dyadic rational has not one, but two distinct representations in the Cantor set.
The doubled representations hold in general: for any given finite-length initial sequence
corresponds to the non-periodic part of the orbit, after which iteration settles down to all zeros (equivalently, all-ones).
It is not hard to see that such repeating sequences correspond to rational numbers.
Writing one then clearly has Tacking on the initial non-repeating sequence, one clearly has a rational number.
In fact, every rational number can be expressed in this way: an initial "random" sequence, followed by a cycling repeat.
For example, geodesics on compact manifolds can have periodic orbits that behave in this way.
That is, imagine sprinkling some dust on the unit interval; it is denser in some places than in others.
Viewed as a linear operator, the most obvious and pressing question is: what is its spectrum?
The uniform density is, in fact, nothing other than the invariant measure of the dyadic transformation.
in greater detail, one must first limit oneself to a suitable space of functions (on the unit interval) to work with.
[7] A vast amount of simplification results if one instead works with the Cantor space
By adjoining set-complements, it can be extended to a Borel space, that is, a sigma algebra.
are a finite number of specific bit-values scattered in the infinite bit-string.
In this case, the operator has a discrete spectrum, and the eigenfunctions are (curiously) the Bernoulli polynomials!
In this case, one finds a continuous spectrum, consisting of the unit disk on the complex plane.
A complete basis can be given in other ways, as well; they may be written in terms of the Hurwitz zeta function.
This then leads elegantly into the theory of elliptic equations and modular forms.
be the inverse temperature for the system, the partition function for this model is given by We can implement the renormalization group by integrating out every other spin.
