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A Solution to Schroeder's Equation in Several Variables | | Robert A. Bridges
; | | Date: |
17 Jun 2011 | | Abstract: | Let phi be a self-map of B^n, the unit ball in C^n, fixing 0, and having
full-rank at 0. If phi (0)= 0, Koenigs proved in 1884 that in the well- known
case n = 1, Schroeder’s equation, f circ phi = phi ’(0) f has a solution f,
which is bijective near 0 precisely when phi ’(0)
eq 0. In 2003, Cowen and
MacCluer formulated the analogous problem in C^n (for a non-negative integer n)
by defining Schroeder’s equation in several variables as F circ phi = phi
’(0)F and giving appropriate assumptions on phi . The 2003 Cowen and MacCluer
paper also provides necessary and sufficient conditions for an analytic
solution, F taking values in C^n and having full-rank near 0 under the
additional assumption that phi ’(0) is diagonalizable. The main result of this
paper gives necessary and sufficient conditions for a Schroeder solution F
which has full rank near 0 without the added assumption of diagonalizability.
More generally, it is proven in this paper that the functional equation F circ
phi = phi ’(0)^k F with k a positive integer, is always solvable with an F
whose component functions are linearly independent, but if k > 1 any such F
cannot be injective near 0. In 2007 Enoch provides many theorems giving formal
power series solutions to Schroeder’s equation in several variables. It is also
proved in this note that any formal power series solution indeed represents an
analytic function on the ball. | | Source: | arXiv, 1106.3370 | | Services: | Forum | Review | PDF | Favorites | |
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