§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 C° ° 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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