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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 
38 
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Fic. 17 
Fic. 13 
A 5. INFERIORITY OF INDEFINITELY FLUCTUATING PATHS 
IF ¢=<0 
We define the asymptotic range of the path (x,, z,) as the 
nonempty closed interval 
(47) [5 3], {=lim inf z,, {= lim sup z, 
— Too t>T Too t=>7T 
A positive length T - ¢ of the asymptotic range implies that z, 
continues to fluctuate between any neighborhood of ¢ and any 
neighborhood of €, infinitely often, and for arbitrarily large ¢. 
LEMMA 2: If p<o and if §<Z for the attainable path (x,, z,), 
then there exists for each N>o an attainable path (x*, z*) and 
a Tx>o0 such that 
1 7-73 
[ 
Urle)= | ee (u(x,) — u(æt)) dt < — N for all T>T, . 
4] Koopmans - pag. 42