Bernoulli figure

Bernoulli shift based on two different starting values and ${\displaystyle 0<x_{01}<0{,}5}$ ${\displaystyle 0{,}5<x_{02}<1}$

The Bernoulli mapping (or Bernoulli shift) is defined as a one - dimensional, time - discrete dynamic system with a piece-wise-defined system function by the rule with the parameter . ${\displaystyle x_{n+1}=ax_{n}{\bmod {1}}}$ ${\displaystyle a>0}$

The Bernoulli map provides interesting properties for. The iteration rule is obtained , i.e. for and for . ${\displaystyle a=2}$ ${\displaystyle x_{n+1}=2x_{n}{\bmod {1}}}$ ${\displaystyle 2x_{n}}$ ${\displaystyle x_{n}<0{,}5}$ ${\displaystyle 2x_{n}-1}$ ${\displaystyle x_{n}>0{,}5}$

The Bernoulli map is chaotic .

The following iteration values ​​are obtained with the start value : ${\displaystyle x_{0}=0{,}4}$

	Decimal system	Binary system
${\displaystyle x_{0}}$	0.4	0.01100110
${\displaystyle x_{1}}$	0.8	0.11001100
${\displaystyle x_{2}}$	0.6	0.10011001
${\displaystyle x_{3}}$	0.2	0.00110011

At this point it becomes clear why the Bernoulli mapping is also referred to as the Bernoulli shift: the binary digit is shifted to the left and the place in front of the decimal point is cut off. Ie after each iteration step the system forgets exactly one digit of the binary representation, ergo one bit of information is lost.

The binary representation also clearly shows that the Bernoulli mapping has several invariant sets when setting the parameters . ${\displaystyle a=2}$

• All rational initial values, the binary representation of which is finite, lead to the orbit landing at the fixed point after a finite number of steps . ${\displaystyle {\hat {x}}=0}$
• All rational initial values, whose binary representation is periodic, lead to the orbit landing on a periodic attractor after a finite number of steps .
• All irrational initial values ​​have an infinite and aperiodic binary representation and therefore form an aperiodic attractor .