CHAPTER 1
What is Single Orbit Dynamics
The study of dynamical systems has its origins in the classical mechanics of
Newton and his successors. The modern theory has flowered in several different
directions that are best described by abstracting certain features of the classical
systems. The ergodic theorem of J. von Neumann pointed the way to the func-
tional analytic - operator ergodic theory while the seminal ergodic theorem of
G.D. Birkhoff founded what we now call measure ergodic theory which can be
viewed as the study of transformations of measure spaces. Keeping only the topo-
logical structure led to the topological dynamics of Gottschalk and Hedlund and
focusing on the analytic or smooth structures gave rise to complex dynamics
and smooth dynamics. Each of the above has expanded to a broad and fairly
independent discipline with its own problems and methodology. Less well known
specialties of a similar nature are generic dynamics where first category, in the
sense of Baire, replaces the /x-null sets as the negligible sets, and measurable or
Borel dynamics where no sets are neglected a priori and the only structure that
comes into play is the Borel structure.
Single orbit dynamics is not another kind of dynamics of the type described
above. It is rather an attempt to focus attention on a rather large body of work
concerned with dynamical study of single orbits as opposed to the global study of
a system as a whole. One can see the origins of this point of view in the works of
R. von Mises, H. Bohr and N. Wiener none of whom actually thought about what
they were doing in the terms that I just used. Nonetheless, I will begin by telling
something about the relevant work of each of these mathematicians before I try to
give some general formulation of single orbit dynamics.
1
http://dx.doi.org/10.1090/cbms/095/01
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