Unformatted text preview: ed points of appropriate compositions of the Tj . Speciﬁcally,
T1 (T3 (T2 (ξ1 , η1 ))) = (ξ1 , η1 )
T2 (T1 (T3 (ξ2 , η2 ))) = (ξ2 , η2 )
T3 (T2 (T1 (ξ3 , η3 ))) = (ξ3 , η3 )
Solving these equations gives the coordinates. For instance, (ξ1 , η1 ) is the solution of
= (ξ1 , η1 )
So (ξ1 , η1 ) = (1/7, 2/7). Similar calculations give (ξ2 , η2 ) = (4/7, 1/7) and
(ξ3 , η3 ) = (2/7, 4/7).
In summary, applying T1 , T2 , and T3 repeatedly in this pattern produces a
sequence of points converging to three points (ξ1 , η1 ), (ξ2 , η2 ), and (ξ3 , η3 ), that
constitute a 3-cycle under these transformations in this order. The coordinates
of these points can be found by noting each (ξi , ηi ) is a ﬁxed point for an
appropriate composition of Ti . This approach of converting periodic points to
ﬁxed points of a more complicated transformation will be of use in our study of
This example generalizes in the obvious way, after...
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