[Article] Chaos: a new mathematical paradigm by Vernon Howe

By Vernon Howe

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Since F maps cr1 onto cr2 , for some r1 , r2 > 0, we have F = ν ◦ g, where ν is a real dilation in zn , and g ∈ AutCR (cr1 ). Analogously to the previous case, the maps f and f1 transform the group G(M )|O(x) into the groups RBn−1 ×S 1 |cr1 and RBn−1 ×S 1 |cr2 , respectively, hence the element g lies in the normalizer of RBn−1 ×S 1 in AutCR (cr1 ). 14) and obtain that au1 = au2 for all u1 , u2 . Furthermore, we have eic µ−1 (u) ≡ µ−1 (e±ic u). Differentiating this identity with respect to c at c = 0 we see that µ−1 (u) = eit u±1 , for some t ∈ R.

At the same time, each of these domains has an n2 -dimensional automorphism group. 1) eiβ zn zn → , (cz + d)2θ where Ab c d ∈ SUn−1,1 , β ∈ R. In this chapter we obtain a complete classification of hyperbolic manifolds with d(M ) = n2 without any additional assumptions. We will now state our main result. 1. ([I2]) Let M be a connected non-homogeneous hyperbolic manifold of dimension n ≥ 2 with d(M ) = n2 . 2) (v) Ar,θ := (z , zn ) ∈ Cn−1 × C : r exp θ|z |2 < |zn | < exp θ|z |2 , with either θ = 1, 0 < r < 1, or θ = −1, r = 0; (vi) Br,θ := (z , zn ) ∈ Cn−1 × C : |z | < 1, r(1 − |z |2 )θ < exp (Re zn ) < (1 − |z |2 )θ , with either θ = 1, 0 ≤ r < 1 or θ = −1, r = 0; (vii) C := (z , zn ) ∈ Cn−1 × C : −1 + |z |2 < Re zn < |z |2 .

3) that dim g0 (D2 ) ≤ n2 − 4n + 6 (note that in this case C = {(x, y) ∈ R2 : x > 0, y > 0}). Further, the explicit descriptions of g1/2 (D2 ) and g1 (D2 ) (see [Sa]) imply that both these components are trivial. Therefore d(D2 ) ≤ n2 − 2n + 4 < n2 − 1, 2 hence D cannot be equivalent to D2 (alternatively, to exclude D2 we could observe that it is not homogeneous). Thus, we have shown that if d(M ) is either n2 + 1 or n2 + 2, then M is holomorphically equivalent to Bn−1 × ∆, which is listed in (i) of the theorem.

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