  We consider "Thurston maps": branched self-coverings of the sphere with
ultimately periodic critical points, and prove that the Thurston equivalence
problem between them (continuous deformation of maps along with their critical
orbits) is decidable.
  More precisely, we consider the action of mapping class groups, by pre- and
post-composition, on branched coverings, and encode them algebraically as
"mapping class bisets". We show how the mapping class biset of maps preserving
a multicurve decomposes into mapping class bisets of smaller complexity, called
"small mapping class bisets".
  We phrase the decision problem of Thurston equivalence between branched
self-coverings of the sphere in terms of the conjugacy and centralizer problems
in a mapping class biset. Our decomposition results on mapping class bisets
reduce these decision problems to small mapping class bisets; they correspond
to rational maps, homeomorphisms and maps double covered by a torus
endomorphism, and their conjugacy and centralizer problems are solvable
respectively in terms of complex analysis, group theory and linear algebra.
  Branched coverings themselves are also encoded into bisets, with actions of
the fundamental groups. We characterize those bisets that arise from branched
coverings between topological spheres, and extend this correspondence to maps
between spheres with multicurves, whose algebraic counterparts are "sphere
trees of bisets".
  To illustrate the difference between Thurston maps and homeomorphisms, we
produce a Thurston map with infinitely generated centralizer --- while
centralizers of homeomorphisms are always finitely generated.
