Breadcrumb
Properties of self-similar sets
Supervisor: Thomas Jordan
Theme: Ergodic Theory and Dynamical Systems
A self-similar set is given by the attractor of an iterated function system consisting of a finite number of contracting similarities. It is well know that if the system satisfies the open set condition then the Hausdorff dimension is given by the solution to $\sum r_i^s=1$ (where $r_i$ are the contraction ratios). However in some cases this still gives the dimension even when the open set condition is not satisfied. A potential problem is to find a condition about the similarities which is equivalent to the Hausdorff dimension dropping below $s$. A further area of study could be when is it possible for self-similar sets with positive Lebesgue measure to have empty interior. In, \cite{CJPPS} it is shown that such sets can exist in $\R^d$ for $d\geq 2$. However whether such sets exist in $1$-dimension is still unknown. A more tractable problem may be to look at the same problem for random self-similar sets in $1$ dimension.
Publications
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Positive measure self-similar sets without interior (2006)
M. Csornyei, T. Jordan, M. Pollicott, D. Preiss and B. Solomyak
Ergod. Th. Dyn. Sys., vol: 26, Issue: no. 3, Pages: 755 - 758
URL provided by the author