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 was once Hayashibara discussion board '92: overseas Symposium on New Bases for Engineering technology, 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 a foreign country participated. They contain either natural and utilized mathematicians of varied backgrounds and represented eleven coun attempts. The organizing committee consisted of the subsequent household individuals and Mike KEANE from Delft: Masayosi HATA, Shunji ITO, Yuji ITO, Teturo KAMAE (chairman), Hitoshi NAKADA, Satoshi TAKAHASHI, Yoichiro TAKAHASHI, Masaya YAMAGUTI the second was once held on the learn Institute for Mathematical technology at Kyoto college from November 30 to December 2 with emphasis on natural mathematical aspect within 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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P is ensured. e. W E fl, 3 nw such that 2 < Wn < an - 2 V n > nw. f. Clearly Kn(W)q2nT - - O. 82n«T("n(w)+1)Q2n whn) - fi2n(W2n) = fi2n(1) = -. n 1 We use the fact that V y > 0, N ~ 1, 3 N < nk(N) Now, for fixed w, y, and N > Too 3 1 L - - = y. :J(w). 10=1 It follows that QNT n --+ 0 whence T Q N U(L2(m)) --+ Id. On the other hand, Thus C(cp) ::) IR+ With some minor adjustments, C(cp) = IR can be arranged. 0 §2 Homogeneous Banach spaces and Koksma inequalities Definition. By a pseudo-homogeneous Banach space on 'JI' we mean a Banach space (B, II· liB) satisfying (1) B ~ Ll('JI'), and II· liB ~ 1I·liI, (2) if fEB and t E 'JI' then ft E B, and IIftllB = IIfIlB, where ft(x) = f(x-t),x E 'JI'.

6. If Tn E IN and E~=I ~. < 00, then {kAn : n ~ 1, 1 :5 k :5 Tn}' C D(cp). Proof. e. e. wEn, ,,,(k~·qn~) ~ T ~ k" Anil ~ a a •e , o and a E D(cp). 6 facilitate easy constructions of conservative, ergodic, coalescent, non-squashable G-extensions of odometers. 7. There is a continuous lR-valued cocycle of product type which is a coboundary, and satisfies Gp(C(cp» = lR. Proof. Assume that E~=I aln < +00, an ~ 3. Let 00 cp(w) = L(bn(Tw) - bn(w» n=I 33 where, as before, bn(w) = fin(wn). Set fi2n+l == 0, and fi2n(k) = {~ o k = 1, else.

O Remarks (1) If T is an invertible, ergodic probability preserving transformation and 'P an ergodic co cycle , and Q(x,g) = (Sx,F(x,g)) is non-singular, and commutes with T"" then Q has the above form. (2) If w: G -+ G is non-singular and measurable, then w is continuous, and onto. To see this, note that w( G) is a mG-measurable subgroup of G, whence :3 X ~ w(G) ::::} xw(G) c G\ w(G) ::::} m(w(G)) = m(xw(G)) ~ m(G \ w(G)) = o. g. G = iZ k X Q' X IRm) then any ergodic G-extension of a Kronecker transformation is coalescent.