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)

(where ${\displaystyle [0,1)^{\infty }}$ is the set of sequences from ${\displaystyle [0,1)}$) 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. This map readily generalizes to several others. An important one is the beta transformation, defined as ${\displaystyle T_{\beta }(x)=\beta x{\bmod {1}}}$. This map has been extensively studied by many authors. 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.

The map can be obtained as a homomorphism on the Bernoulli process. Let ${\displaystyle \Omega =\{H,T\}^{\mathbb {N} }}$ be the set of all semi-infinite strings of the letters ${\displaystyle H}$ and ${\displaystyle T}$. These can be understood to be the flips of a coin, coming up heads or tails. Equivalently, one can write ${\displaystyle \Omega =\{0,1\}^{\mathbb {N} }}$ the space of all (semi-)infinite strings of binary bits. The word "infinite" is qualified with "semi-", as one can also define a different space ${\displaystyle \{0,1\}^{\mathbb {Z} }}$ consisting of all doubly-infinite (double-ended) strings; this will lead to the Baker's map. The qualification "semi-" is dropped below.

This space has a natural shift operation, given by

where ${\displaystyle (b_{0},b_{1},\dots )}$ is an infinite string of binary digits. Given such a string, write