In recent years, geometric measure theory has become a very heated topic in mathematics. According to Herbert Federer in , “[the advances of geometric measure theory] have given us deeper perception of the analytic and topological foundations of geometry, and have provided new direction to the calculus of variations”. In other words, a good understanding of the theory is essential to the study of modern mathematics.
In Real Analysis II, we have learned some foundations of basic Measure Theory. In this independent studies project, we include a further investigation of Measure Theory and the application of it in fractal geometry.
In Chapter 1, some basic notions of measure theory, such as σ-algebra, Borel sets, measurable sets are included as a foundation to later chapters. In Chapter 2, we will specifically introduce Lebesgue measure, Hausdorff measure, Hausdorff dimension, and their properties. Lastly, in Chapter 3, we will focus closely on Julia sets, visualize the connection between Hausdorff dimension and complex dynamics, and eventually state some main results from Shishikura’s paper , which talks about the Hausdorff dimension of the boundary of Julia sets. Due to the complexity of the proofs in the last section of Chapter 3, readers who are especially interested shall refer to more advanced literatures, and the original proof is not going to be duplicated in this project.
Xu, Ruifeng, "On Hausdorff Dimension and Julia Sets" (2018). Senior Independent Study Theses. Paper 8082.
Hausdorff dimension, Hausdorff dimension of Julia sets, Julia sets, Hausdorff measure
Bachelor of Arts
Senior Independent Study Thesis
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