A Study Of Dimension Theory Based On A Class Of Self-affine Sets And Self-affine Measures

Given a metric space, one of the most important ways of constructing fractals is via iterated function systems(IFS). At first, people primarily concern with self-similar sets, which are a special kind of the attractors of IFS. As time goes on, the dimension theory of self-similar sets is gradually improving. Some people turned to consider a more complicated IFS fractal- selfaffine set. In Euclidean space, self-affine sets generalize self-similar sets.This thesis will discuss the dimension theory of self-affine sets. Comparing with selfsimilar sets, which only need to consider the size and position of its covers, various shapes of the covers of self-affine sets are a notoriously difficult point. This thesis will split the study of self-affine sets into two parts: the generic case and the specific case.1) The generic case will be discussed in Chapter 5, which focuses on studying the relationship between the dimension of self-affine sets and the transitions of its IFS. According to de Finetti’s theorem, we genealize the result given by Jordan, Pollicott, Simon [23].2) The specific case will be discussed in Chapter 3 and Chapter 4. In Chapter 4, we will mainly concentrate on calculating the Minkowski dimension of parts of Feng-Wang sets and then assert that the dimension can be found by a formula. One can show that it contains the result given by Feng, Wang [15]. Moreover, we will study counterexamples that cannot apply this result. The first one is a Barański set, and so we can apply Barański’s result [2]and conclude that the Minkowski dimension still satisfies the formula. The second one is much more complicated. We apply the combinatorial result given by Feng, Wang [15] and prove that the dimension still satisfies the formula. Finally, we provide a lower bound for its Hausdorff dimension by using Mc Mullen’s method.