TWIST PERIODIC ORBITS FOR CONTINUOUS MAPS OF THE EIGHT SPACE

. Let g be a continuous map from 8 to itself has a fixed point at (0,0) ,we prove that g has a twist periodic orbit if there is a rational rotation number

Abstract. Let g be a continuous map from 8 to itself has a fixed point at (0,0) ,we prove that g has a twist periodic orbit if there is a rational rotation number

Introduction:
In [1], Alseda,L. et.al introduced the definition of twist periodic orbit, they studied this type of periodic on continuous circle map of degree one which has a fixed point In. [4], ] Misurewicz,M studied the twist sets for circle maps. Let 8 be the one point union of two unit circles attached at (0,0) and , be a map has a fixed point at (0,0), we say that z 8 is a periodic point of g if there exists a positive integer n such that The period of z is the smallest integer satisfying this relation. Let be the set of periods of if z 8 is a periodic point of period n ,then the orbit of z is the set Throughout this work , we will use e as projection from s 1 onto 8 .There exists many projections from s 1 into 8 , but we define e as: This projection is one to one .We can find a lift map f from s 1 to itself such that eg =fe Let G be a lift to g and a projection of R 8. It is clear  is many to one, G is not defined uniquely ; that is if G 1 and G are two lifting of g then G= G 1 + m with m  Z. the degree of g is the number a which satisfy . In this work, we use a=1. If z 8 s a periodic point of g of period n and then this imply is the rotation number of z we denoted it by We denote by the set of all rotation numbers of is called the rotation set of g .Since f is unique lift to g then the properties of rotation number and rotation set on the circle satisfy on the eight space. So the rotation set does not depend on the choice of t , but it depends on the periodic orbit . Also, if is a convergent sequence of a and The another property If g has no periodic point the rotation number exist for all and it is independent of t and is irrational number there are more properties on circle maps satisfy on 8 (see [ Services and Standards Online: 2013-06-03 ISSN: 2277-8020, Vol. 6, pp 4-8 doi:10.18052/www.scipress.com/BSMaSS.6.4 2013 Then notion of TPO of period m characterizes the simplest behavior of the graph a map which has the same rotation number.

Main Theorems
In this section , we prove two theorems which find the TPO, also we prove other properties of it. Since, F on is one to one and by definition of TPO order preserving then G(t i )=t i+j ; for some j  2 in the same way t i +d =G m (t i )=G m-1 (t i+j )=...=t i+mj =t i +j(by claim 2)thus d=j; where j  2 this imply g(z i )=f(0)=0, so z i is an eventually fixed point.
An A-graph of g with respect to A m and B m is an oriented graph with vertices I 1 ,I 2 ,…, I m+n-1 such that if I a f covers I  k times but not k+1 times then there are k arrows from I a to I  . a sequence is an A-graph of g is called a path of length r and the path is called a loop if a loop is called simple if The lemma below generalize lemma in[2] on circle map: Lemma 1.2.5: Let be a continuous map of degree one. If is a loop in a A-graph of g then there exists a fixed point z of g n such that Lemma 1.2.6: Let g be an monotone map (not necessarily continuous) of a closed interval I into itself. Then g has a fixed point.

Proof: suppose g is increasing map Let
since g is increasing, if but this contraction.
We call the A linearization of G. we denote by the map of 8 of degree one which has as lift , also we say the A linearization of g . If that is then we say that G and g are linear. We claim X is a TPO, to show that :Since periodic orbit on that is Now we will compare between and X. let such that a set is to left and which to the right. The elements of the U-partition are :Case 1: If these intervals are different then are the end points of U-partition, therefore we get X is a TPO. Case 2: If they are the same , then one has to go along the orbits of these two points, since we only use increasing pieces of the maps G and then in the same order, since T is a TPO so X is a TPO.
We will define these maps as: let and G be a lift of g and We call such that G r is a lift of g r and such that G l is a lift of g l this mean g r, g l are continuous maps of degree one.
Proof : Without loss of generality,If we assume g is linear on W and F is increasing on then this don't for generality.Then by lemma 1.2.7, let w be a periodic orbit of g of period m and let U be the partition of 8 by elements of W ,since g is onto then for all