In 1992 two successive symposia were held in Japan on algorithms, fractals and dynamical systems. The first one was Hayashibara Forum '92: International Symposium on New Bases for Engineering Science, Algorithms, Dynamics and Fractals held at Fujisaki Institute of Hayashibara Biochemical Laboratories, Inc. in Okayama during November 23-28 in which 49 mathematicians including 19 from abroad participated. They include both pure and applied mathematicians of diversified backgrounds and represented 11 countries. The organizing committee consisted of the following domestic members and Mike KEANE from Delft: Masayosi HATA, Shunji ITO, Yuji ITO, Teturo KAMAE (chairman), Hitoshi NAKADA, Satoshi TAKAHASHI, Yoichiro TAKAHASHI, Masaya YAMAGUTI The second one was held at the Research Institute for Mathematical Science at Kyoto University from November 30 to December 2 with emphasis on pure mathematical side in which more than 80 mathematicians participated. This volume is a partial record of the stimulating exchange of ideas and discussions which took place in these two symposia.