Student(s):
Marshall Carpenter
Dates of Involvement:
Fall 2008-Spring 2009
Faculty Advisor(s):
James Curry and Anne Dougherty
Graduate Mentor:
Finding the Hausdorff Dimension of a Self-Affine Set
Introduction:
Fractal dimensions can be easily found for iterated function systems which involve similarity contractions, but very little work has been done with affine contractions. The fixed point of an iterated function system which employs affine transformations is termed a self-affine set. An affine contraction distorts the image and creates a smaller version which is not similar, but has undergone a linear transformation and a translation. The problem is to find bounds on the Hausdorff dimension of a self-affine fractal set generated by several different iterative methods. The same set can be created by a recursive iterative process using the de-Rahm Chaikin algorithm and it can also be produced as the result of an iterated function system with affine contractions.
Results:
An entire family of fractals can be created by changing the parameters of the iterated function system and of the de-Rahm Chaikin algorithm. Numerical approximations can be made for the dimension of each of these fractals, but there is no analytic proof that their dimension is greater than 1. The current goal of this research is to analytically find a lower bound the dimension of these self-affine sets. Once a mechanism is in place, it may be possible to generalize an algorithm to find the dimension of iterated function systems with affine contractions.
What is very interesting about this formulation is that we have four different systems all converging to the same fixed point. What is interesting here is that the left column is a collection of lines which crosses over themselves, the two center columns are a collection of points, and the right column is a collection of areas.
A method has been found which bounds the dimension of these sets strictly less than two. The lower bound is often much more complicated to find and it is doubtful that an exact measure of the dimension would be found. The first priority is to bound the dimension above one and then more research may continue to find an exact value or to generalize the system.
About Marshall Carpenter:
Marshall Carpenter is currently in his second year at CU-Boulder, looking to complete a BS/MS program in Applied Math. His primary interests are dynamical systems with simulations and complex variables. He hopes to pursue a doctorate degree and become a teacher of mathematics or chemistry.
