Contemporary Mathematics
Volume 215, 1998
Robert Ellis and the Algebra of Dynamical Systems
H. Furstenberg and E. Glasner
1. Introduction
From its inception dynamical systems theory was a branch of analysis, with the
mathematical counterpart of physical dynamical systems taking the form of systems
of ordinary differential equations, or vector fields on manifolds, In the hands of
Poincare and Birkhoff, the topological character of dynamical systems theory came
to light and one development of this was the establishment of topological dynamics
as a branch of dynamical systems theory.
It
is Robert Ellis who perceived the
profound role that algebra was to play in the development of topological dynamics,
and who shaped the instruments that proved to be powerful tools in this area.
To be sure, the setting was different from the classical one, with compactness
being a crucial hypothesis, and the focus being more on "factor systems" than
subsystems (e.g. periodic orbits, asymptotic cycles). The reason that the focus
shifted to factors rather than subsystems is that a considerable part of the theory
is devoted to studying minimal systems which do not have subsystems. While it is
true that no "ergodic decomposition" exists to analyze a general (compact) system
in terms of minimal ones, the minimal systems are in some sense still the basic
building blocks, paralleling the role of simple groups in group theory.
Undoubtedly one of the inspirations for Ellis' ideas was Mackey's notion of
"virtual" subgroup. Ergodicity is an approximation to transitivity of a group action,
and while a strictly transitive action of a group
T
on a space can be represented
by the action on a quotient T / A, A a subgroup of T, we can regard an ergodic
action as coming from a "virtual subgroup"
A
of
T.
This idea has borne fruit in
ergodic theory, but it has never had more than heuristic value, because in interesting
situations virtual subgroups had no real counterparts, What Ellis showed was that
in topological dynamics one could attach an actual group to a compact dynamical
system and that a considerable part of the dynamics was reflected in the group
and its relation to other groups. This idea would be appropriate for any point-
transitive system, the analogue of an ergodic system, namely a system with a dense
orbit (and in the metric case, a residual set of dense orbits). For such systems the
dynamics "lives" on the acting group T, Namely if T acts on the compact space
X, (t,x)
-+
tx, then X is determined by C(X) which is faithfully represented on
the dense orbit Txo with f
E
C(X) corresponding to fo(t)
=
f(txo), the action
of t'
E
T being determined by the algebra automorphism, fo(t)
t--
fo(tt'). For
simplicity we shall assume that
T
is discrete, We wish to associate to aT-invariant
1991 Mathematics Subject Classifications:
Primary 54H20.
©
1998 American Mathematical Society
3
http://dx.doi.org/10.1090/conm/215/02926
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