  We prove that the algorithm of [13] for approximating the Hausdorff dimension
of dynamically defined Cantor sets, using periodic points of the underlying
dynamical system, can be used to establish completely rigorous high accuracy
bounds on the dimension. The effectiveness of these rigorous estimates is
illustrated for Cantor sets consisting of continued fraction expansions with
restricted digits. For example the Hausdorff dimension of the set $E_2$ (of
those reals whose continued fraction expansion only contains digits 1 and 2)
can be rigorously approximated, with an accuracy of over 100 decimal places,
using points of period up to 25.
  The method for establishing rigorous dimension bounds involves the
holomorphic extension of mappings associated to the allowed continued fraction
digits, an appropriate disc which is contracted by these mappings, and an
associated transfer operator acting on the Hilbert Hardy space of analytic
functions on this disc. We introduce methods for rigorously bounding the
approximation numbers for the transfer operators, showing that this leads to
effective estimates on the Taylor coefficients of the associated determinant,
and hence to explicit bounds on the Hausdorff dimension.
