  Following van der Poorten, we consider a family of nonlinear maps which are
generated from the continued fraction expansion of a function on a
hyperelliptic curve of genus $\mathrm{g}$. Using the connection with the
classical theory of J-fractions and orthogonal polynomials, we show that in the
simplest case $\mathrm{g}=1$ this provides a straightforward derivation of
Hankel determinant formulae for the terms of a general Somos-4 sequence, which
were found in a particular form by Chang, Hu and Xin, We extend these formulae
to the higher genus case, and prove that generic Hankel determinants in genus
two satisfy a Somos-8 relation. Moreover, for all $\mathrm{g}$ we show that the
iteration for the continued fraction expansion is equivalent to a discrete Lax
pair with a natural Poisson structure, and the associated nonlinear map is a
discrete integrable system. This paper is dedicated to the memory of Jon Nimmo.
