I first began studying ergodic theory in my master's project during the summer of 1993. Since January of 1995 I have been working with Andres del Junco on a Ph.D. thesis. I have been investigating problems in the area of joinings. I shall graduate in 1999. Please select some of the following links to learn more about ergodic theory and the people I work with.
| What is Ergodic Theory? | My Academic Family Tree | Preprints |
| Ergodic Theory Conferences | Joinings in Ergodic Theory | Current Research |
In simple terms Ergodic theory is the study of long term averages of dynamical systems. OK, maybe that's not simple to everyone. A dynamical system is anything that changes (with time). Mathematicians study very complicated dynamical systems like our earth. It changes quite a lot with time and this can be hard to express or quantify mathematically. We work also on very simple dynamical systems, for example, an object moving at a constant speed in a perfect circle around a fixed point. Ergodic Theory asks the question: what is the average position of the object?
The kind of Ergodic Theory I do is concerned with measure preserving
transformations (m.p.t.'s) of a Lebesgue space (i.e. the unit interval
[0,1] with the usual lebesgue measure). The central problem here is to
identify all such transformations. Properties of transformations that are
invariant under isomorphism include ergodicity, weak mixing, mixing, and
entropy. By using these invariants and others Ergodic theorists will some
day be able to, if not identify, then classify all m.p.t.'s.
| Upcoming Conferences | |
| AMS
Southeastern Section Meeting
University of Florida in Gainsville March 12-13, 1999 |
I speaking at this conference on a problem from my thesis
"A measure preserving transformation with square roots of all orders but no infinite square root chain." |
| Past Conferences | |
| New
York Journal of Mathematics Conference
SUNY Albany June 9-13, 1997 |
This was an excellent week long conference. See the conference proceedings. |
| Kai Lai Chung | Ray Chacon | Mustafa Ackoglu (1963)
(Unofficial student of Ray Chacon. Actual supervisor was R.D. Kodis in electrical engineering). |
J.R. Baxter (1969) | Lawrence Susanka (1991) |
| Philip Jones (1995) | ||||
| Jie Liao (1996) | ||||
| Joseph Cunsolo (1969) | ||||
| Andres del Junco (1974) | Reem Yassawi (1998) | |||
| Blair Madore
(1999 - In progress) |
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| Marcus Pivato (in progress) | ||||
| Peter Fong (in progress) | ||||
| Hugh Miller (1978) | ||||
| Dogan Çömez (1983) | ||||
| Daniel Boivin (1986) | ||||
| Chris Bose (1986) | ||||
| Felix Lee (1988) | ||||
| Rob Bradley (1989) | ||||
| David McIntosh | ||||
| Xian Wei Ha (1994) | ||||
| M. Dzung Ha | ||||
| Sebastien Ferrando (1994) | ||||
| Nat Friedman | ||||
| Tom Schwarzbauer | ||||
| Steve McGraf | ||||
| Jim Olsen | ||||
| Neil Falkner | ||||
| John Walsh | ||||
| Naresh Jain | ||||
| Elton Hsu | ||||
Given a transformation T on a lebesgue space X with borel sigma algebra and measure u. Can you find a measure m on the product space X x X so that the T x T will be a measure preserving transformation for m? If so then you've found a self joining of T. In general a joining of (X,T,u) to (Y,S,v) is a measure m on X x Y that is invariant under T x S.
The simplest self joining of (X,T,u) is the diagonal measure, D. This measure is defined by D(A) = u({x | (x,x) in A }), i.e. the measure D is just u projected onto the graph of f(x)=x. You can also define a joining by projecting u onto the graph of any measure preserving transformation S that commutes with T. That measure is often denoted (I x S)D meaning D((I x S)^(-1)(A)) for any set A in the product sigma algebra. It's not hard to check this measure is invariant under T x S.
To see how this connects to my current research you may want to look
at my research statement or my
CV.
I am finalizing a paper (from my thesis) and will make it available here soon.
1. A Z^2 rank 1 mixing action where all times are simple.