Fractional iteration of functions of two variables

Phil Diamond
1970 Bulletin of the Australian Mathematical Society  
Let X € {R 2 , C 2 } and T • «i = h v1 x k y l , Vl = lb v1 x k y l (fc, Z » 0 , k+l > l ) be an invertible holomorphic function from a neighbourhood of 0 € if to ? 10 ? 0 1 the linear part of T . In this thesis, 010 b 01 ) the fractional iteration of T is examined when (1) X = C 2 , * = [ j 0] , 0 < |»| 5 |a| < 1 , (2) X = C 2 , A = [« i] , 0 < |a| < 1 , (3) X = R 2 , X = f° j] , b 2 < ha , 0 < a < 1 , (10 x = R 2 . A-In the first three cases the fractional iterates are obtained from the
more » ... ined from the algorithm T(s) = lim f o B 8 o f 1 where B is the linear or almost linear part of a suitable normal form of T . In case h, an asymptotic expansion is obtained for the natural iterates "f 1 of higher order near the fixpoint 0 , and this leads to an algorithmic solution of the functional equations \(Tx) = A(x) -1 , \i{Tx) = u U ) , x € X .
doi:10.1017/s000497270004171x fatcat:tpljhxvxbzbsjks7uapypdlzau