# Mathematics Colloquium

#### Thursday, December 5, 1996, 4:00pm

#### 102 Bradley Hall

## Professor Frantisek Franek

#### McMaster University

speaks on

### Structural Properties of Universal Minimal Dynamical
Systems for Discrete Semigroups

**Abstract.**Newton solved the 2-body problem by explicit integration of the underlying
differential equations. This approach did not work for the general n-body problem. In 1890's
Poincare' brought the geometry of the phase space (space of "possible values") to bear on the
analysis to investigate the integral curves in their entire domain of existence (Poincare'
reccurence theorem). Thus, the focus shiffted away from differential equations that define a
dynamical system to the phase space and the group of transformations implicit in the system.
Birkhoff (1920's) discussed many dynamical phenomena in the context of transformation
groups acting on general metric spaces. Poincare's approach gave rise to a local theory of
dynamical systems, while Birkhoff's approach gave rise to an abstract theory of dynamical
systems, also called Topological Dynamics, where the action of a general group of
transformations on a phase space is studied. We shall focus on topological dynamical systems
where a discrete semigroup is acting on a compact Hausdorff space. Of course, such acting is not
possible via continuous bijections (as in the case of groups), but simple continuous self-maps.
As a consequence such dynamical systems may be "wild" and hard to classify and describe.
Nevertheless, a special subclass of such dynamical systems, so-called "universal minimal"
systems, are much more conducive to classification and description. In a bit of a surprise, there
are strong connections to the theory of Boolean algebras.
We show that for a discrete semigroup S there exists a uniquely determined complete Boolean
algebra B(S) - the algebra of clopen subsets of M(S). M(S) is the phase space of the universal
minimal dynamical system for $S$ and it is an extremally disconnected compact Hausdorff}

space. We deal with this connection of semigroups and complete Boolean algebras focusing on
structural properties of these algebras. We shall present some of the more interesting results
and techniques.

**Tea. **High tea will be served at 3:30pm in the Lounge.

**Emmy's. **Certain refreshments will be available at Emmy's after the talk.

**Host. **Frank Tall is the host. Anybody who is interested in having dinner
with the speaker should contact Frank at 646-2421.