我們會先介紹一些碎形幾何中的一些基礎定義，像是豪斯多夫距離、豪斯多夫維度、疊代函數系統及吸引子，並研究一些碎形理論的基礎結果，如：壓縮映射原理、拼貼定理、箱子計數定理、莫蘭定理，我們會用二維數據建造疊代函數系統，並證明疊代函數系統的係數及碎形維度之間的關係，最後藉由以上結果來進行股價預測。

We introduce some basic notions in fractal geometry such as Hausdorff distance, Hausdorff dimension, iterated function system (IFS) and attractor. Some fundamental results such as the contraction mapping principle, the collage theorem, the boxing counting theorem and the Moran theorem are studied. We construct IFS from some given data sets, and show that the coefficients of the IFS are related to the dimensions of the attractor of the IFS. These results are applied to stock market prediction.

目錄
1. Introduction -1
2. Fractals In Metric Space -3
3. Iterated Function Systems -13
4. Fractals In Metric Measure Space -23
5. Applications In Stock Price Prediction -35
Reference -43

[1] 胡家信(2013)。《分形分析引論》。北京：科學出版社。
[2] Micheal F. Barnsley, Fractals Everywhere (third ed), Dover Publication, New
York, (1993).
[3] Edgar. Gerald A, Measure, Topology, and Fractal Geometry, New York,
(2007).
[4] Barnsley. M.F, Elton. J, Hardin. D, Massopust. P, Hidden Variable Fractal
iterated Function, Georgia Institute of Technology Preprint, July 1896, to
appear in SIAM Journal of Analysis.
[5] Kedong. Yin, Hengda. Zhang, Wenbo. Zhang, Qian. Wei, Fractal Analysis of
the Gold Market in China, Journal for Economic Forecasting, Institute for
Economic Forecasting, Vol. 0(3), 144-163, October.
[6] John E.Hutchinson, Fractals And Self Similarity, Indiana University Mathe-
matics Journal 30 (1981),713-747.
[7] Halsey Royden, Patrick M. Fitzpatrick, Real Analysis (Fourth ed), Pearson
Education Asis Limited and China Machine Press, (2010).
[8] D,LA Torre, F. Mendivil, The Monge-Kantovich Metric On Multi-measures
And Self-similar Multi-measures, Mathematics Subject Classication, (2014),
28B,28A33,60B10,28A80.
[9] Rudin, Walter, Principles of mathematical analysis, McGraw-Hill Science En-
gineering, (1976).
[10] D. P.Hardin, P. R. Massopust, The capacity of a class of fractal function,
Common math. Phys, (1986), 455-460.