220 IV Continued Fractions and Their Uses
of the origin and f(z)=λz+O(z^2 ),whereλ =e^2 πiθfor some irrationalθ.It
is readily shown that there exists a formal power serieshwhich linearizes f,i.e.
f(h(z))=h(λz). Brjuno (1971) proved that this formal power series converges in
a neighbourhood of the origin if
∑
n≥ 0 (logqn+^1 )/qn<∞,whereqnis the denomi-
nator of then-th convergent ofθ. It was shown by Yoccoz (1995) that this condition
is also necessary. In fact, if
∑
n≥ 0 (logqn+^1 )/qn=∞, the conclusion fails even for
f(z)=λz( 1 −z). See Yoccoz [62] and P ́erez-Marco [40].
Our discussion of continued fractions has neglected their analytic theory. The out-
standing work of Stieltjes (1894) on theproblem of moments, which was extended by
Hamburger (1920) and R. Nevanlinna (1922) from the half-line to the whole line, not
only gave birth to the Stieltjes integral but also contributed to the development of func-
tional analysis. For modern accounts, see Akhiezer [2], Landau [33] and Simon [55].
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