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The dyadic transformation (also known as the dyadic map, bit shift map, 2x mod 1 map, Bernoulli map, doubling map or sawtooth map) is the mapping (i.e., recurrence relation) 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. [1]

## Almost all

In mathematics, the term "almost all" means "all but a negligible amount".

## Bernoulli polynomials

In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, occur in the study of many special functions and, in particular the Riemann zeta function and the Hurwitz zeta function.

## Bernoulli process

In probability and statistics, a Bernoulli process is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1.

## Bernoulli scheme

In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes.

## Binary number

In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: typically 0 (zero) and 1 (one).

## Bitwise operation

In digital computer programming, a bitwise operation operates on one or more bit patterns or binary numerals at the level of their individual bits.

## Blancmange curve

In mathematics, the blancmange curve is a fractal curve constructible by midpoint subdivision.

## Chaos theory

Chaos theory is a branch of mathematics focusing on the behavior of dynamical systems that are highly sensitive to initial conditions.

## Complex quadratic polynomial

A complex quadratic polynomial is a quadratic polynomial whose coefficients and variable are complex numbers.

## Continuous spectrum

In physics, a continuous spectrum usually means a set of attainable values for some physical quantity (such as energy or wavelength) that is best described as an interval of real numbers, as opposed to a discrete spectrum, a set of attainable values that is discrete in the mathematical sense, where there is a positive gap between each value and the next one.

## Eigenfunction

In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue.

## Gilbert–Shannon–Reeds model

In the mathematics of shuffling playing cards, the Gilbert–Shannon–Reeds model is a probability distribution on riffle shuffle permutations that has been reported to be a good match for experimentally observed outcomes of human shuffling, and that forms the basis for a recommendation that a deck of cards should be riffled seven times in order to thoroughly randomize it.

## Groupoid

In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways.

## Hurwitz zeta function

In mathematics, the Hurwitz zeta function, named after Adolf Hurwitz, is one of the many zeta functions.

## Integrable system

In the context of differential equations to integrate an equation means to solve it from initial conditions.

## Iterated function

In mathematics, an iterated function is a function (that is, a function from some set to itself) which is obtained by composing another function with itself a certain number of times.

## List of chaotic maps

In mathematics, a chaotic map is a map (.

## Logistic map

The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations.

## Map (mathematics)

In mathematics, the term mapping, sometimes shortened to map, refers to either a function, often with some sort of special structure, or a morphism in category theory, which generalizes the idea of a function.

## Modular group

In mathematics, the modular group is the projective special linear group PSL(2,Z) of 2 x 2 matrices with integer coefficients and unit determinant.

## Orbit (dynamics)

In mathematics, in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system.

## Piecewise linear function

In mathematics, a piecewise linear function is a real-valued function defined on the real numbers or a segment thereof, whose graph is composed of straight-line sections.

## Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

## Recurrence relation

In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given: each further term of the sequence or array is defined as a function of the preceding terms.

## Spectrum (functional analysis)

In mathematics, particularly in functional analysis, the spectrum of a bounded operator is a generalisation of the set of eigenvalues of a matrix.

## Square-integrable function

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

## Tent map

In mathematics, the tent map with parameter μ is the real-valued function fμ defined by the name being due to the tent-like shape of the graph of fμ.

## Topological conjugacy

In mathematics, two functions are said to be topologically conjugate to one another if there exists a homeomorphism that will conjugate the one into the other.

## 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.

## Turn (geometry)

A turn is a unit of plane angle measurement equal to 2pi radians, 360 degrees or 400 gradians.

## References

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