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More on the Minimal Set of Periods
إضافات لأصغر مجموعة دورية

Author: Afraa R. Sadek عفراء راضي صادق
Journal: Iraqi Journal of Science المجلة العراقية للعلوم ISSN: 00672904/23121637 Year: 2015 Volume: 56 Issue: 4A Pages: 2953-2958
Publisher: Baghdad University جامعة بغداد

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Abstract

Let M be a n-dimensional manifold. A C1- map f : M  M is called transversal if for all m N the graph of fm intersect transversally the diagonal of MM at each point (x,x) such that x is fixed point of fm. We study the minimal set of periods of f(M per (f)), where M has the same homology of the complex projective space and the real projective space. For maps of degree one we study the more general case of (M per (f)) for the class of continuous self-maps, where M has the same homology of the n-dimensional sphere.

ليكن M منطوي ذو بعد n . الدالة المستمرة والقابلة للاشتقاق f: M  M تسمى مستعرضة اذا كان لكل  N m, بيان الدالة fm يقطع قطر MM بصورة مستعرضة لكل نقطة (x,x), حيث ان x نقطة صامدة الى fm. ستقوم بدراسة اصغر مجموعة دورية الى (M per (f)) f, حيث ان M يمتلك نفس همولوجي المستوى الاسقاطي العقدي والمستوى الاسقاطي الحقيقي, بالنسبة للدوال ذات الدرجة واحد سندرس حالة اعم لحساب (M per (f)) لمجموعة الدوال الذاتية المستمرة, حيث ان M يمتلك نفس همولوجي الكراة ذات البعد n .


Article
the coincidence lefschetz number for self-maps of lie groups
عدد لبشز للتطابق الدوال من زمرة لي الى نفسها

Authors: Ban Jaffar AL-Ta'iy بان جعفر حسن حياوي الطائي --- Adil G.Naoum عادل غسان نعوم
Journal: Baghdad Science Journal مجلة بغداد للعلوم ISSN: 20788665 24117986 Year: 2006 Volume: 3 Issue: 3 Pages: 465-469
Publisher: Baghdad University جامعة بغداد

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Abstract

Let/. It :0 ---0 G be any two self maps of a compact connected oriented Lie group G. In this paper, for each positive integer k , we associate an integer with fk,hi . We relate this number with Lefschetz coincidence number. We deduce that for any two differentiable maps f, there exists a positive integer k such that k 5.2+1 , and there is a point x C G such that ft (x) = (x) , where A is the rank of G . Introduction Let G be an n-dimensional com -pact connected Lie group with multip-lication p ( .e 44:0 xG--+G such that p ( x , y) = x.y ) and unit e . Let [G, G] be the set of homotopy classes of maps G G . Given two maps f , f G ---• Jollowing [3], we write f. f 'to denote the map G-.Gdefined by 01.11® =A/WO= fiat® ,sea Given a point g EC and a differ-entiable map F: G G , write GA to denote the tangent space of G at g [4,p.10] , and denote by d x F the linear map rig F :Tx0 T, (x)G induced by F , it is called the differential of Fat g [4,p.22]. Let LA, Rx :0 G be respec-tively the left translation Lx(i)=4..(g,e) , and the right translation Rx(1)./..(gcg). Then there is a natural homomorphism Ad ,the adjoin representation, from G to GL(G•), (the group of nonsingular linear transformations of Qdefined as follows:- Ad(g)= deRe, od,Lx. Note that d xRc, ad.; =d(4,( Lx(e)))0 de; =d.(4, 04)=4(40 Re) = d(4(4, (e)))0 (44, =d ar, o (44, . Since G is connected , the image of Ad belongs to the connected component of G(G)containing the identity,i.e. for each g E 0, detAd(g) > 0 . By Exercise Al • Dr.-Prof.-Department of Mathematics- College of Science- University of Baghdad. •• Dr.-Department of Mathematics- College of Science for Woman- University of Baghdad.

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