タイトル：Fractal geometry and related topics
日時：2023/7/27
講演者:伊縫 寛治（慶応大学 研究員）
会場：武蔵野研究開発センタ
Since the importance of fractals was recognized by Benoit Mandelbrot in 1970s, fractals have been studied.
Indeed, fractal geometry arises in many contexts and is applied to not only physics but also other fields now.
Note that, fractal geometry is a theory to obtain geometrical properties of "irregular" sets (fractals) and one of major topics in fractal geometry is to estimate the "size" of fractals (dimensions and measures of fractals).
In this talk, we first explain the motivations for fractal geometry, and consider typical theorems and examples of fractals generated by iterated function systems (for short, IFSs) to estimate the dimensions and measures of the fractals.
Note that these fractals are "ideal" in the mathematical sense and therefore there are rich mathematical results on the estimation of the dimensions and the measures of the fractals.
We finally consider a generalization of fractals generated by IFSs and consider a example of the generalization introduced by D. Mauldin and M. Urbanski (1996).
Note that this example has a connection to the continued fractions in number theory.
If we have a time, we explain their result for the example of the generalization, and related results proved by the speaker and the collaborators.
This study is a joint work with Hiroki Sumi (Kyoto University) and Hikaru Okada (Osaka University).