  In the late 1980's Sullivan initiated a programme to prove quasisymmetric
rigidity in one-dimensional dynamics: interval or circle maps that are
topologically conjugate are quasisymmetrically conjugate (provided some obvious
necessary assumptions are satisfied). The aim of this paper is to conclude this
programme in a natural class of $C^3$ mappings. Examples of such rigidity were
established previously, but not, for example, for real polynomials with
non-real critical points.
  Our results are also new for analytic mappings. The main new ingredients of
the proof in the real analytic case are (i) the existence of infinitely many
(complex) domains associated to its complex analytic extension so that these
domains and their ranges are compatible, (ii) a methodology for showing that
combinatorially equivalent complex box mappings are qc conjugate, (iii) a
methodology for constructing qc conjugacies in the presence of parabolic
periodic points.
  For a $C^3$ mapping, the dilatation of a high iterate of any complex
extension of the real map will in general be unbounded. To deal with this, we
introduce dynamically defined $qc\backslash bg$ $partitions$, where the
appropriate mapping has bounded quasiconformal dilatation, except on sets with
"bounded geometry". To obtain such a partition we prove that we have very good
geometric control for infinitely many dynamically defined domains. Some of
these results are new even for real polynomials, and in fact an important
sequence of domains turn out to be quasidiscs. This technology also gives a new
method for dealing with the infinitely renormalizable case.
  We will briefly also discuss why quasisymmetric rigidity is such a useful
property in one-dimensional dynamics.
