MAT 401: Seminar in Mathematics (Hausdorff dimesion and fractals)

Fall 2004

Christopher Bishop

Professor, Mathematics
SUNY Stony Brook

Office: 4-112 Mathematics Building
Phone: (516)-632-8274
Dept. Phone: (516)-632-8290
FAX: (516)-632-7631

The Hausdorff dimension of a set is a number which mesures how big the set is in a certain sense. It generalizes the usual notion of dimension in the sense that points have diemsnion 0, a line segment has dimension 1, a surface has dimension 2 and so on. However, there are many sets which have non-integer dimensions. For example, the famous ``Cantor middle thirds set'' is a compact set in the line which has dimension log 2/ log 3. The ``von Koch Snowflake'' is a curve in the plane which is homeomorphic to a circle, but has dimension log 4/ log 3.

In this seminar we will discuss the basic definitions and properties of dimension and spend most of our time computing dimensions of various sets. For example, what is the dimension of the set of real numbers whose binary expansions have twice as many zeros as ones (on average)? What is the dimension of the graph of a random function?

To get the most out of the course you should have had some analysis (and measure theory would be a big help, but not necessary). Terms like compact set, open, closed, convergent sequence, ... should be familar to you. We will also be using measures quite a bit so if you have seem some measure theory that would be good, but I will plan to summarize what we need as we go along. (A measure is a function from sets to non-negative numnbers that satisfies certain reasonable conditions, e.g., the measure of a disjoint union is the sum of the individual measures. Mapping an interal to its length is an example of a measure; seeing how to extend the definition to all sets is where measure theory comes in.)

We will use some lecture notes of Yuval Peres and myself as a basis for the course. I will post the chapters as postscript files on the webpage. We may also have some other handouts as time, interest and needs may be.

The course will be graded based on class participation (presenting a part of the book in class) and problem sets. I will post a specific point/grade scheme once we have met and I have seen how big the class is and the level of preparation of the students. all.pdf (Chapters 1-4 of the book by Bishop and Peres). This is a fairly big file (around 40M, or 130 pages) so you may want to download it, but only print pages you really want. There are a few more chapters; I may post these later.

Link to history of mathematics There are a lot of iteresting articles here. If you know of other math related sites I should link to, let me know.

Assigmnent 0 (Due Tuesday, Sept 7): Prove that the Minkowski dimension of the unit interval [0,1], exists and is equal to 1.

Assigmnent 1 (Due Tuesday, Sept 14):

-------- Problem 1: Suppose $K$ is a set in Eucildean space, $\epsilon >0$ and $N(K,\epsilon) \leq C_1 \epsilon^{-\alpha} + C_2$. Show the Minkowski dimension of $K$ is $\leq \alpha$. Suppose $N(K,\epsilon) \geq A_1 \epsilon^{-\beta} - A_2$. Show the Minkowski dimension is $\geq \beta$.

-----------Problem 2: Prove that the Minkowski dimension of the unit interval [0,1], exists and is equal to 1.

-----------Problem 3: Compute the Minkowski dimension of the middle thirds Cantor set.

Assigmnent 2 (Due Tuesday, Sept 21):

-------- Problem 1: Suppose $K$ is the countable set formed by the union of $0$ and the sequence $n^\alpha$. Compute the Minkowski dimension of $K$.

-----------Problem 2: Prove that the upper Minkowski dimension of a subset of [0,1] equals the upper Minkowski dimension of its closure.

-----------Problem 3: Suppose $E_0,E_1, E_2,...$ is a decreasing, nested sequence of sets in [0,1]. Let $E$ be the intersection of these sets. Is the upper Minkowski dimension of $E$ equal to the limit of the diemnsions of the $E_n$'s? Find a proof or a counterexample.

Assigmnent 3 (Due Tuesday, Sept 28):

-------- Problem 1: Show the Hausdorff dimension of a countable set is zero.

-------- Problem 2: Show that for subsets of the unit interval, Hausdorff dimension is bounded above by lower Minkowski dimension.

-------- Problem 3: Find a set K whose Hausdorff dimension is 0, but whose Minkowski dimension is positive.

Assigmnent 4 (Due Tuesday, Oct 12):

-------- Problem 1: Suppose S is the set of integers which is the union of blocks [2n!, (2n+1)!]. Show the supper density is 1 and the lower density is 0.

-------- Problem 2: For the set A_S defined in class (and the notes) show the lower minkowski dimension is same as the lower density of S and the upper Minkowski dimension is the same as the upper density of S

-------- Problem 3: Find a set whose upper and lower Minkowski dimensions are different.