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Then Cco(z)c CPCO(z). Since e E Y C X ( ~ E Owe ) , have Cco(z)C CPco(z) C X \ K and so z E X(EO). Since f ( Y )= Y , we have nm(p-l)co(~)) cn cY 00 Yc 00 m(Eo)) i=O i=O nzo and thus Y = fi(N(p-l)co(Y)). Since C ( p - l ) c o ( zC) Cclco(z) C X \ K for C x ( , - l ) c 0 ( f ( ~and ) ) thus f(N(P--l)CO L E Y , we have f ( C ( P - l ) c o ( z3) ) ( Y ) )3 N x ( P - l ) c o ( y ) 3 ~(jA-1)co(Y). Therefore, Y = N(P-l)co(Y) and Y is open in X. This is a contradiction. 21. Consider the subset X in the plane defined by x = {% : I%[ 3 1 = 1) u {% : 1% - -1 = -} 2 2 u {% : 1% + -132 1 = -}.

This is followed by defining a Lyapunovfunction V((zl, y l ) , ( X 2 , Y Z ) ) by = -(Y2 - Y l ) ( Y 2 - 2 2 - ( Y l - x1)). 2 Anosov differentiable systems 25 In fact, V : T2 x T2 4 R is continuous, V(z',z')= 0 for z' E T2 and V(g(z'), g(y')) -V(z',y') > 0 whenever 0 < d(z',y') 5 a for a > 0 small enough. Then g is expansive. This is easily checked as follows. Let 0 < e < a. Then e is an expansive constant for 9. Indeed, suppose 3: # y and d(g"(z),g"(y)) 5 e for all n E %. Since T2 is compact, there exists 6 > 0 such that for u,v E T2 ).

N;=, u,"==, In the remainder of this section we describe well known theorems that will be used in the sequel. Let X be a topological space. A path in X is a continuous map from the unit interval [0,1] to X. If any two points in X are joined by a path, then X is said to be path connected. In general, a connected space need not be path connected. An arc in X is an injective continuous map from [0,1] to X. We say that X is arcwise connected if any two points in X are joined by an arc. It is clear that if X is arcwise connected then it is path connected.

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A Beckman-Quarles type theorem for finite desarguesian planes by Benz W.

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