Differentiable and Complex Dynamics of Several Variables by Pei-Chu Hu

By Pei-Chu Hu

The improvement of dynamics thought all started with the paintings of Isaac Newton. In his concept the main simple legislation of classical mechanics is f = ma, which describes the movement n in IR. of some degree of mass m lower than the motion of a strength f by means of giving the acceleration a. If n the location of the purpose is taken to be some extent x E IR. , and if the strength f is meant to be a functionality of x purely, Newton's legislation is an outline by way of a second-order usual differential equation: J2x m dt = f(x). 2 It is smart to minimize the equations to first order through defining the velo urban as an additional n self sufficient variable by means of v = :i; = ~~ E IR. . Then x = v, mv = f(x). L. Euler, J. L. Lagrange and others studied mechanics by way of an analytical procedure referred to as analytical dynamics. at any time when the strength f is represented through a gradient vector box f = - \lU of the capability power U, and denotes the adaptation of the kinetic power and the aptitude strength by means of 1 L(x,v) = 2'm(v,v) - U(x), the Newton equation of movement is decreased to the Euler-Lagrange equation ~~ are used because the variables, the Euler-Lagrange equation might be If the momenta y written as . 8L y= 8x' additional, W. R.

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Dg(zn, wn ) n-+oo Since K is compact and by the uc-normal family hypothesis, without loss of generality we may suppose that the sequence {In} converges uniformly to a holomorphic mapping f E Hol(M, N) on any compact sub set of M, and that Zn -+ Zo, Wn -+ Wo for some Zo, Wo E K, as n -+ 00. If Wo =F Zo, then lim dh (fn (Zn) , fn(wn)) = dh(f(ZO) , I(wo)) dg(zn,w n ) n-+oo < +00, dg(zo, WO) contradicting OUf assumption. Thus we have Wo = Z00 Note that M is locally compact. We can choose relatively compact neighborhood U of Zo such that U is contained in a local co ordinate neighborhood.

Here we extend the Makienko's conjecture (cf. 6 Assume that M is a compact oriented smooth rn-dimensional mani/old and / E C(M, M) satis/ying 1) Ideg(f)I ~ 2, and 2) F(f) has no components which are backward invariant, then ResJ(f) =I- 0. S. 31 Let / be a rational /unction 0/ degree at least two. Then 1) i/ResJ(f) is not empty, ResJ(f) is a completely invariant subset 0/ J(f) and dense in J(f), moreover, it contains uncountably many points; 2) i/ J(f) is disconnected and i/ there exists no completely invariant component 0/ F(f), ResJ(f) is not empty.

Assume Jpoi(f) = 0. Then for each x E M, there is a positive number p = p(x) such that M(x; p) is a Poincare recurrent domain. Set En(x) = {y E M(x;p) I {j(y),j2(y), ... ,r(y)}nM(x;p) = 0}, Un(x) = M(x;p) -En(x). Then Un(x) C UM1 (x) for n ? 1. Since M(x; p) is a Poincare recurrent domain, we have fL(M(x; p)) = J-t (U n=l Un(x)) = n-+oo lim fL(Un(X)), which implies lim fL(En(x)) = O. n-+oo Hence for every positive number c, there exists an integer N(x) such that when n ? N(x) J-t(En(x)) < c. Note that M is compact and note that {M(X;P)}XEM is an open covering of M.

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