§1. Introduction
W. Thurston's pseudo-Anosov maps, although central to his
classification of diffeomorphisms of compact surfaces up to isotopy
([T], [F-L-P]), are themselves homeomorphisms which are diffeomorphisms
except at finitely many points (singularities) where they fail to
be differentiable. (On the torus a pseudo-Anosov map is an Anosov
diffeomorphism, but for surfaces of genus greater than one, singu-
larities must occur.) In [G-K], a construction was given that
associated to each pseudo-Anosov map f, a C diffeormophism g
which is topologically conjugate to f through a homeomorphism
isotopic to the identity. For surfaces of genus greater than one,
the topological conjugacy must be of a global nature, because f
cannot be made a diffeomorphism by a coordinate change that is
smooth outside the singularities (or even outside a sufficiently
small neighborhood of the singularities). (See [G-K], paragraph
2.4.)
Pseudo-Anosov maps have interesting dynamical properties
which are shared with the smooth models of [G-K]. They minimize
both the number of periodic points (for every period) and the
topological entropy in their isotopy classes. Moreover, a
pseudo-Anosov map is Bernoulli with respect to a natural absolutely
continuous invariant measure whose density is C00 and positive
except at the singularities, where it vanishes. The smooth models
of [G-K] are actually Bernoulli with respect to smooth measures
(i.e. ones whose densities are positive and ° in every local
coordinate system).
Research partially supported by NSF Grant No. MCS-8202055.
Received by the editors November 7, 1983; and in revised form July 11, 1984.
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