  We study a class of homeomorphisms of surfaces collectively known as
linked-twist maps. We introduce an abstract definition which enables us to give
a precise characterisation of a property observed by other authors, namely that
such maps fall into one of two classes termed co- and counter-twisting. We
single out three specific linked-twist maps, one each on the two-torus, in the
plane and on the two-sphere and for each prove a theorem concerning its ergodic
properties with respect to the invariant Lebesgue measure.
  For the map on the torus we prove that there is an invariant, zero-measure
Cantor set on which the dynamics are topologically conjugate to a full shift on
the space of symbol sequences. Such features are commonly known as topological
horseshoes. For the map in the plane we prove that there is a set of full
measure on which the dynamics are measure-theoretically isomorphic to a full
shift on the space of symbol sequences. This is commonly known as the Bernoulli
property and verifies, under certain conditions, a conjecture of Wojtkowski's.
We introduce the map on the sphere and prove that it too has the Bernoulli
property.
  We conclude with some conjectures, drawn from our experience, concerning how
one might extend the results we have for specific linked-twist maps to the
abstract linked-twist maps we have defined.
