FractionalGeometry-Chap2

43 and so for any sequence i1 i2 ik diamtik ti1

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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 111 1 1 111 , = (ξ1 , η1 ) ξ1 + η1 + 222 2 222 2 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 chaos. This example generalizes in the obvious way, after...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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