Fractal Geometry, Dynamical systems and number theory.

分形几何和动力系统中的许多问题都使用数论的技术。我已经列出了三种这样的问题。自相似集的拓扑。设f1， f2，…， fn是线性收缩的有限集合。存在一个唯一的非空紧集K，使得K = f1(K)并集…工会fn (K)。缩并的集合称为迭代函数系统，K是相关的自相似集。这些迭代函数系统的一个简单特例是当我们只有两个映射时，从R2到R2， f1(x) = A x + b和f2(x) = A x - b。有许多有趣的问题很难回答。例如，在A上需要什么条件才能使所得到的自相似集合连通、完全不连通或包含非平凡的内部？自相似测度的维数如果我们赋予迭代函数系统中的线性收缩非零概率，我们就构造了自相似测度。对于多重分形分析、局部维数以及这些测度在底层线性收缩满足开集条件下的结构，我们已经有了很多了解。这是一个确保fi(K)和fj(K)没有任何非平凡重叠的条件。在有明显重叠的情况下，我们知道的要少得多。一种情况是，自相似测度满足有限型条件。具有有限类型条件的自相似测度具有嵌入到相关自相似测度构造中的非常精确的组合结构。这样的组合结构允许应用一般的计算技术。最后一个有趣的问题通常被归类为“有洞的地图”。考虑一个理想的台球游戏，在一张固定形状的桌子上玩，桌子上的某个地方有一个洞。对于这个台球游戏，我们现在考虑一个台球，从一个随机的位置开始，朝着一个随机的方向运动。人们可能会问的一些典型问题是：台球掉进洞里的概率是多少，或者永远避开洞的概率是多少？如果球掉入洞中，到它掉入洞的预期时间是多少？避免孔洞的随机位置/方向的集合有可能具有测度0，但可能仍然具有正的豪斯多夫维数。关于这个维度我们能说些什么呢？我们现在看到的场景更加抽象，处理的是离散动力系统，而不是台球游戏中的连续动力系统，但潜在的问题仍然是一样的。这有助于阐明哪些结果是特定于连续系统的，哪些是具有更广泛影响的更一般的结果。

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.