What is a Dynamical System?

0, 1, 2 counting problem. In the past Research section I described "A chaotic, non-mixing subshift". A neat counting question that comes out of the paper is: what is the entropy of the system? Entropy is a quantity which can be used to measure just how chaotic a system is; I conjectured that the answer is "not very chaotic"--that the entropy is only log(2).

The question can be translated into this form: Suppose we write down a number with n digits, and this number satisfies two criteria:

• The digits in the number can only be 0, 1, or 2, and
• we can only put 2's in the spots which are not a power of 2 away from a 0.

That is, "010112" is an acceptable number with n=6 digits because the '2' is 3 and 5 spaces away from the zeroes, but "100112" is not, because there are a '0' and a '2' exactly 4 spaces apart.

The question: Approximately how many acceptable numbers are there of length n?

The question has been investigated by a number of mathematicians. Art Benjamin of Harvey Mudd carried the problem with him to the Spring Combinatorics Seminar (New Orleans, 1996). Combinatorics is the mathematics most concerned with counting strategies; he and his colleagues did not manage to solve it. The U.S. Military Academy made this problem one of the main topics of investigation of their summer mathematics seminar during 1997; they used their super-computer to find exact answers up to n=38 (there are more than 2,000,000,000,000 acceptable numbers with 38 digits). In the spring of 1997, Joe Auslander (from the University of Maryland) carried the problem to a Dynamics seminar in Israel. Two of the luminaries of measure-theoretic dynamics--Dan Rudolph and Benjamin Weiss--worked together to solve the problem, and showed that my conjecture is correct: the entropy of the system is log(2) (or, the acceptable numbers of length n grow approximately in proportion to 2ⁿ ).

The solution is beautiful and surprising for a number of reasons. The first is that it uses measure-theoretic methods rather than combinatorial methods (a fact which will not impress the general reader, but is of considerable surprise to mathematicians). The second is that the argument used to prove the result is not a standard one; it uses many deep and unrelated properties of dynamics in a new way. But possibly most surprising is that the problem, which seems like such a standard mathematical question, does not yield to standard arguments.

Quasicontinuous Functions. I now return to the baker's transformation, which I introduced at the beginning of these pages as a typical example in the field of dynamics. It is an especially useful example to measure-theoretic dynamicists, who appreciate the uniform slope. Topological dynamicists, however, are forced to treat the baker's transformation as a "special case" because topologists deal almost exclusively with continuous functions--which the baker's transformation is not. Building a bridge between the theories of these two fields, therefore, means finding a way to bring the baker's transformation and other functions like it into a well-understood (from the topological point of view) class of functions. We need to describe a class of discontinuous--but not too badly discontinuous--functions for which standard topological theorems still hold.

In our paper "Dynamics of Quasi-Continuous Systems", Mario Martelli and I attempt to do just that. Martelli presented two definitions of chaos which he proved are equivalent for continuous functions, and asked how far we can push the equivalence. My contribution was to describe a class of functions (which we call "quasi-continuous" or "semicontinuous") where this equivalence still holds. To our great delight, quasi-continuity has proved to be useful far beyond Martelli's theorem. In fact, we show that many of the most famous and beautiful results of continuous topological dynamics hold for my quasi-continuous systems as well.

A quasicontinous function f:X-->Y has the property that if V is an open set in Y, then the inverse image of V in X is semi-open.

A set A is semi-open if the closure of the interior of A contains A: that is A lies in cl(int(A)).

After that paper is done, there are a plethora of questions I still want to know the answer to:

• Nice quasi-continuous functions look like the baker's transformation. What do "bizarre" quasi-continuous functions look like?

• Must the set of points of continuity of a quasi-continuous function have positive measure?

• How can we use quasi-continuous functions to make the connections between measure-theoretic dynamics and topological dynamics more explicit?

• What are the limitations of this class: which topological theorems do not hold, and why?

Return to Crannell's home page