Differential Equations: Conference Proceedings by Ian W. Knowles, Roger T. Lewis

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By Ian W. Knowles, Roger T. Lewis

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By positive G-multiplicative solutbrs of Pu = 0 in M. In particular when M is of parabolic type (that is when is abelian) the results described in the previous the fundamental group a,(M) sections can be extended to the (lifted) operator P in 2 where M is any non-compact covering space of M. 1. Under the assumption that G is abelian, suppose that the equation Pu = 0 admits a positive solution in- M. Then the equation also admits a G-multiplicative positive solution in M. Similarly when G is abelian one can establish a representation theorem for positive solutions of Pu = 0 in M which generalizes the representation theorem given at the end of section 2.

U. Afikakos and R. Rostamian and p a s s i n g t o z e r o set s Ao(x) = 6(x) = D i r a c ' s d e l t a f u n c t i o n . The s-average i s d e f i n e d as t h e c o n s t a n t ( i f it e x i s t s ) f o r which t h e f o l l o w i n g a,(s) convergence (S - rN-S u o ( r x ) AVD) : + aD(s)As(x) D'(lRN). h o l d s i n t h e s e n s e of d i s t r i b u t i o n s a,(s) and a,(s) = aB(s), as r If + e x i s t s , t h e n so does a,(s) but t h e converse i s n o t t r u e , [ l ] . For t h e f u t u r e r e f e r e n c e , l e t ' s i n t r o d u c e h e r e one f i n a l d e f i n i t i o n .

F o r i # j . I f z1 # t h e k - c y c l e i s s a i d t o be a t t r a c J t i v e , i n d i f f e r e n t o r r e p u l s i v e a c c o r d i n g as l F ( k ) ( z l ) l < 1, =1 o r >1 respecF ( z k ) = z,, tively. If z1 = l e t G(z) = l / F ( l / z ) and use G ( k ) ( 0 ) i n p l a c e o f F ( k ) ( z l ) t h e p r e v i o u s sentence. The J u l i a set J o f F ( z ) can be d e f i n e d as t h e s e t of p o i n t s z such t h a t z belongs t o a r e p u l s i v e k - c y c l e of F ( z ) f o r some k E in M.

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