  We develop a new duality between endomorphisms of measure spaces, on the one
hand, and a certain family of positive operators, called transfer operators,
acting in spaces of measurable functions on, on the other. A framework of
standard Borel spaces is adopted; and this generality is wide enough to cover a
host of applications. While the mathematical structures of positive operators,
endomorphisms, transfer operators, measurable partitions, and Markov processes
arise in a host of settings, both pure and applied, we propose here a unified
study. This is the general setting of dynamics in Borel measure spaces. Hence
the corresponding linear structures are infinite-dimensional. Nonetheless, we
prove a number of analogues of the more familiar finite-dimensional settings,
for example, the Perron-Frobenius theorem for positive matrices, and the
corresponding Markov chains. Tools from the theory of operators in Hilbert
space of special significance to us will be the use of a certain universal
Hilbert space, as well as classes of operators in it, directly related to the
central theme of duality for transfer operators. From ergodic theory, we
address such questions as measurable cross sections, partitions, and Rohlin
analysis of endomorphisms of measure spaces. While there are classical theorems
dealing with analogous questions for automorphisms of measure spaces, a
systematic study of endomorphisms is of more recent vintage;-- in its infancy.
In order to make the exposition accessible to students and to researchers in
neighboring areas, we have included a number of explicit examples and
applications.
