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.

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http://dx.doi.org/10.1090/cbms/095/01