By Jon Aaronson, Toshihiro Hamachi, Klaus Schmidt (auth.), Y. Takahashi (eds.)

In 1992 successive symposia have been held in Japan on algorithms, fractals and dynamical structures. the 1st one used to be Hayashibara discussion board '92: foreign Symposium on New Bases for Engineering technological know-how, Algorithms, Dynamics and Fractals held at Fujisaki Institute of Hayashibara Biochemical Laboratories, Inc. in Okayama in the course of November 23-28 within which forty nine mathematicians together with 19 from in another country participated. They comprise either natural and utilized mathematicians of diverse backgrounds and represented eleven counĀ­ attempts. The organizing committee consisted of the next family contributors and Mike KEANE from Delft: Masayosi HATA, Shunji ITO, Yuji ITO, Teturo KAMAE (chairman), Hitoshi NAKADA, Satoshi TAKAHASHI, Yoichiro TAKAHASHI, Masaya YAMAGUTI the second used to be held on the examine Institute for Mathematical technological know-how at Kyoto collage from November 30 to December 2 with emphasis on natural mathematical facet during which greater than eighty mathematicians participated. This quantity is a partial checklist of the stimulating alternate of rules and discussions which happened in those symposia.

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Suppose that f: 7l' --+ JR, J~ f(t) dt = 0 is piecewise linear, and Ef=1 dj =I 0, then for each irrational number a the corresponding cylinder flow T, is ergodic. Proof. There is no loss of generality in assuming that Ef=1 dj > Riemann integrable, the ergodic theorem holds uniformly, so 1 q-l -q j=o L f'(x + ja) --+ 10 1 0 o. Since f' IS f'(t) dt > 0 uniformly in x. 1) On the other hand, f(q) is still piecewise linear with the discontinuity points of the form Xi + ja, with the jump at it equal to di , where i = 1, ...

Let r be the spiral defined by its polar equation p= f(O) and suppose furthermore that for large 0, r is locally convex. Then its dimension and the critical inverse tempemture f3c are linked by the formula ~ = max{I,f3c} In particular, if r has infinite length then In example 1, f3c = 2/(1 example 3, f3c = o. + a) ~ ~ = f3c. for all a > O.

The difficult part of the proof is the demonstration that Sand T are rationally related when S-normality and T-normality coincide. That is where the hypothesis that ST = T S is used and we conjecture that it may be removed. The 'easy' part of the proof of Theorem 1 depends on the fact that S-normality coincides with Sk-normality. Even in the one-dimensional case this is not entirely obvious and, indeed, the corresponding property is typically violated in the non-integer case which we are about to discuss.

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