# trichotomy

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## tri·chot·o·my

(tri-kot'ŏ-mē),
A division into three parts.
[G. trichia, threefold, + tomē, a cutting]
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(ii) The evolution family U = [{U(t, s)}.sub.t,s[member of]R] is called [beta]-exponentially trichotomic (see [7]) if it satisfies (i) above, except the relation (10) which is replaced by
(i) The evolution family U = [{U (t, s)}.sub.t,s[member of]R] is [alpha]-exponentially trichotomic;
(i) The evolution family U = [{U(t, s)}.sub.t,s[member of]R] is [beta]-exponentially trichotomic;
In this section we show that all the [beta]-exponentially trichotomic evolution families are topologically equivalent to a standard equation with evolution operator
Suppose that the evolution family U is exponentially trichotomic, has bounded growth and decay, and verify in addition relation (13).
(i) If U is [alpha]-exponentially trichotomic, then it is topologically equivalent to the following standard evolution family [V.sub.1]:
(ii) If U is [beta]-exponentially trichotomic, then it is topologically equivalent to the standard evolution family [V.sub.2]:
Furthermore, if we set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we notice that V and W are [alpha], respectively [beta]-exponentially trichotomic, but they are not topologically equivalent to the standard evolution operators [V.sub.1] (t, s) = [e.sup.[absolute value of s]-[absolute value of t]], respectively [W.sub.1](t) = [e.sup.[absolute value of s]-[absolute value of t]] I.
We present a generalization of Theorem 2 in [5] for exponentially trichotomic evolution families:
Suppose that the evolution families U and V are exponentially trichotomic (either a or 8), they both have bounded growth and decay, and satisfy in addition
(P(t), Q(t), R(t), respectively [P.sub.1] (t), [Q.sub.1] (t), R1 (t) are the trichotomic structural projections).
Since the condition (13) seems to be too much restrictive, we need to identify larger classes of trichotomic families for which Theorem 13 is applicable, i.e.
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