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A map is Devaney chaos when:

Write a point as a binary expansionApart from the harmless choice of expansion at dyadic rationals, the binary shift representation of the doubling map is

Every open interval contains a binary cylinder specified by a finite initial word. Given cylinders $U$ and $V$, choose a binary sequence beginning with the word for $U$ and place the word for $V$ after it. A suitable iterate shifts the second word to the front, proving topological transitivity.

Given any cylinder, repeat its defining word forever. The resulting point is periodic and lies in that cylinder, so periodic points are dense.

Finally, given $x$ and any neighbourhood, choose $N$ so large that changing only digits after the first $N$ stays inside that neighbourhood. Choose the later tail so that after $N$ shifts it is either $0$ or $3/4$, whichever is farther from $F^N(x)$. The separation is at least $3/8$, so any smaller fixed constant, for example $\delta =1/4$, proves sensitivity. Hence the doubling map is chaotic in Devaney's sense.

Starting with $x_0=x>0$, the continued-fraction algorithm setsunless $x_n$ is already an integer. This gives

Every finite continued fraction is rational, since it is built from integers by finitely many additions and reciprocals. Conversely, if $x=p/q$ is rational in lowest terms, thenWhen $r\ne 0$, the next complete quotient is $q/r$. Thus each step is an application of the Euclidean algorithm and replaces the denominator by a smaller nonnegative remainder. The process must terminate. This proves the termination criterion for a simple continued fraction.

For $x=\sqrt{3}$,andFinally,so the complete quotients repeat. Hence the continued fraction of the square root of three is