Furstenberg's x2x3 theorem

Furstenberg's x2x3 theorem is a number-theoretical result from the mathematical theory of dynamic systems . It is named after Hillel Furstenberg and is a classic example of the fact that a dynamic system obtained by combining several semigroups can have different properties than the systems generated by the individual semigroups.

Statement of the sentence

For every irrational number there is the quantity ${\displaystyle x\in \mathbb {R} }$

${\displaystyle \left\{2^{m}3^{n}x\mod 1\colon m,n\in \mathbb {N} \right\}}$

tight in . ${\displaystyle \mathbb {R} /\mathbb {Z} }$

Dynamic interpretation

Consider the multiplicative effect of the multiplicative semigroup on the circle . Furstenberg's x2x3 theorem then states that the orbites of irrational numbers are close together. (In contrast, orbits of rational numbers can be periodic .) ${\displaystyle \langle 2,3\rangle :=\left\{2^{m}3^{n}\colon m,n\in \mathbb {N} \right\}}$ ${\displaystyle S^{1}=\mathbb {R} /\mathbb {Z} }$

A stronger statement would be Fürstenberg's x2x3 conjecture. This states that the only -invariant probability measures on the circle are the Lebesgue measure and certain finite linear combinations of Dirac measures . ${\displaystyle \langle 2,3\rangle }$

Furstenberg's x2x3 theorem is also noteworthy because there are numerous Cantor sets (and correspondingly numerous probability measures) that are invariant under the action of one of the two semigroups or (but not under both). ${\displaystyle \left\{2^{m}\colon m\in \mathbb {N} \right\}}$ ${\displaystyle \left\{3^{n}\colon n\in \mathbb {N} \right\}}$

literature

• Furstenberg: Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory 1 (1967) 1-49.

Web links

• Gorodnik: Dynamical Systems in Number Theory (Lecture 10)