Fractal Geometry, Dynamical systems and number theory.

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Many problems in fractal geometry and dynamical systems use techniques from number theory. I have listed three such types of problems.******Topology of self-similar sets.******Let f1, f2, ..., fn be a finite set of linear contractions. There exists a unique non-empty compact set K such that K = f1(K) union ... union fn(K). The set of contractions is called an iterated function system, and K is the associated self-similar set. A simple special case of these iterated function systems is the case when we have only two maps, from R2 to R2, with f1(x) = A x + b and f2(x) = A x - b. There are many interesting questions that are quite difficult to answer. For example, what conditions are needed on A so that the resulting self-similar set is connected, totally disconnected, or contains non-trivial interior? ******Dimensions of self-similar measures******If we assign non-zero probabilities to the linear contractions in an iterated function system, we construct something known as a self-similar measures. A lot is known about the multi-fractal analysis, the local dimension and the structure of these measures in the case where the underlying linear contractions satisfies the open set condition. This is a condition that ensures that fi(K) and fj(K) do not have any non-trivial overlap. In the case where there is significant overlap, much less is known. One situation where things can be said is in the case where the self-similar measure satisfies the finite type condition. Self-similar measures with the finite type condition have a very precise combinatorial structure embedded into the construction of the associated self-similar measure. Such combinatorial structures allow for general computational techniques to be applied. ******Maps with holes******A last question of interest is typically classified as "maps with holes". Consider an idealized game of billiards played on a table of some fixed shape, and a hole somewhere on this table. Given this game of billiards, we now consider a billiard ball, starting from a random location, and traveling in a random direction. Some typical questions one might ask are: What is the probabilities that the billiard ball falls in the hole, or instead avoids the hole forever? If the ball falls into the hole, what is the expected time until it falls into the hole? It is possible that the set of random location/directions that avoid the hole have measure 0, but might still have a positive Hausdorff dimension. What can be said about this dimension? The setting we are looking at is more abstract, dealing with discrete dynamical systems instead of the continuous ones in the billiard game, but the underlying questions remain the same. This helps to shed light on to which results are specific to continuous systems, and which are more general results with wider reaching implications.