Point Off Research Articles

Periodic Points and Contractive Mappings

Let X be a non-empty set and f:X→X. A point x ∈ X is (i) a fixed point off f(x)=x, and (ii) a periodic point of f iff there is a positive integer N such that fN(x)=x. Also a periodic orbit of f is the (finite) set {x, f(x), f2(x),…} where x is a periodic point of f.

Complex iterated radicals

We prove the convergence of the sequence S defined by Z+1 = (Z_-c) 2, c real, for any choice of z,. Let k=j-i_cj2. If c L, S has the fixed points w,=i+ik and w2=1-ik and converges to w1 if Re(zD)>O and to w2 otherwise. We show that convergence is strictly monotone when the neighborhood system is the pencil of coaxial circles with w1 and w2 as limiting points, and give rates of convergence. The purpose of this paper is to prove that the sequence of complex numbers defined by Zn+,=(z-C)12, c real, converges for any choice of z0, i.e. globally, and to discuss the limit and rate of convergence. This problem was posed by C. S. Ogilvy [1]. As a guide to global convergence we first investigate local convergence. The following facts are known about the sequence defined by zn?+=f(zn) for some choice of z0. LEMMA 1. If limrn,, z=w and f is continuous at w then f(w)=w, i.e. w is a fixed point. PROOF. W4=1imn--joo Zn+1=1im.oDf(z.)=f(w)Suppose w is a fixed point off. If ZN=W for some N, then z,,=w for all n_N and we call the sequence trivial. We define q(n, w)=jz,.+j-wj1jzn-w|. LEMMA 2. Suppose w is afixedpoint off andf '(w) exists. If If'(w)I 1, then the sequence cannot converge to w except trivially. Received by the editors June 15, 1972 and, in revised form, February 13, 1973. AMS (MOS) subject classifications (1970). Primary 40A05, 41A25; Secondary 39A10, 50D45.

Closing stable and unstable manifolds on the two sphere

Let f be a diffeomorphism of the two sphere. In this note we prove that if the unstable manifold of a fixed point p forf accumulates on the stable manifold of p, then/can be approximated arbitrarily closely Cr, r> 1, such that they intersect. 1. The results. Let d be a distance on the two sphere S2 coming from a Riemannian metric. The stable inanifold of a point p is defined to be Ws(p,f) = {x S2: d(fnx,fnp) 0 as n o co}. The unstable mnanifold of p is the stable manifold for f-1, Wu(p,f)= WS(p,f-'). Let W`(p,f)-{p}= Ws(p,f)' and Wu(p,f)-{p}= Wu(p,f)'. A fixed point p is called a saddle point if the eigenvalues 2, ,u of the derivative Df(p) satisfy O , and p a fixed saddle point off such that W8(p, f )' nclosure WU(p, f) 0 0 . Then f can be approximated arbitrarily closely Cr byf' such thatf'(p)=p and WS(p,f')' n Wu (p, f )# O. COROLLARY. There is a residual subset (complement of a first category set) R of the set of all Cr diffeomorphisms, Diff r(S2), such that iff e R, p isa saddlefixedpoint off, and Ws(p,f )' nclosure Wu(p,f) $ 0 then Ws(p,f)' r) WU(p,f)$ QS The reason for restricting to S2 is to use the Jordan separation theorem. Therefore analogous results are true on the two disk and R2. In the case of the disk, the above theorem should prove useful to prove a conjecture of Smale in [6]. For this use it would be nice to prove the result assuming only that {p}$closure Ws(p,f) r\closure Wu(p,f). Next, I would hope the results would be true for periodic points but was unable to prove it. There are all sorts of related closing lemma conjectures. See [5]. Received by the editors February 24, 1973. AMS (MOS) subject classifications (1970). Primary 58F10. 1 This research was partially supported by the Organization of American States and the National Science Foundation, GP 19815. (a) American Mathematical Society 1973

Dual model with second-sheet resonances for pion-nucleon scattering

A crossing-symmetric pion-nucleon scattering amplitude with Regge behaviour and second-sheet resonance poles is constructed using suitable combinations of a representation proposed by Cohen-Tannoudji, Henyey, Kane and Zakrzewski (CHKZ). The Regge residues for high-energy scattering are obtained in terms of the functionf(t) which is used in construction of the CHKZ representation. An attempt is made to find the form off(t) which gives the best fit to the high-energy chargeexchange differential cross-section in the forward direction. It is also shown that the nucleon pole term imposes restrictions on the location of the branch point off(t), and that the fit at large |t| can be improved by modifying the π trajectory function so that it satisfies the bound |αρ (t)| < |C|t|1/2 log |t1|| fort→-∞.

The Morse lemma on Banach spaces

Let f : U → R f:U \to R be a C 3 {C^3} map of an open subset U of a Banach space E. Let p ∈ U p \in U be a critical point of f ( d f p = 0 ) f(d{f_p} = 0) . If E is a conjugate space ( E = F ∗ ) (E = {F^ \ast }) we define what it means for p to be nondegenerate. In this case there is a diffeomorphism γ \gamma of a neighborhood of p with a neighborhood of 0 ∈ E , γ ( p ) = 0 0 \in E,\gamma (p) = 0 with \[ f ∘ γ − 1 ( x ) = 1 2 d 2 f p ( x , x ) + f ( p ) . f \circ {\gamma ^{ - 1}}(x) = \frac {1}{2}{d^2}{f_p}(x,x) + f(p). \]