Maps that are roots of power series

Abstract

We introduce a class of polynomial maps that we call polynomial roots of powerseries, and show that automorphisms with this property generate the automorphism group in any dimension. In particular we determine generically which polynomial maps that preserve the origin are roots of powerseries. We study the one-dimensional case in greater depth.

Constant weighted sums of iterates

“…In this article we require that the maps P N n¼0 a n f n converge uniformly to a constant function either in a neighbourhood of some point z or on every compact subset of C k . This of course implies point-wise convergence to 0 of all the derivatives at z (as in (5)), so having a constant weighted sum of iterates near z ¼ 0 implies that f is the root of a power series in Complex Variables and Elliptic Equations 373 the sense discussed in [13]. Having a constant weighted sum of iterates near 0 is in fact strictly stronger.…”

“…In [13] a generalization of locally finite maps was studied by Maubach and the author. Instead of looking at polynomial endomorphisms that are roots of polynomials as in Equation (3), one could consider the maps that are roots of power series: X 1 n¼1 a n f n ¼ 0:…”

“…Having a constant weighted sum of iterates near 0 is in fact strictly stronger. For example, in [13] it was shown that if f is a polynomial and 0 ¼ f(0) is a repelling fixed point, then f is a root of a power series. But if 0 is a repelling fixed point then it lies in the Julia set, and in Corollary 7 we will see that f cannot have a c.w.s.i.…”

“…In [13] point-wise convergence of each coefficient was considered. It was shown that the polynomial automorphisms that are roots of a power series do generate the automorphisms groups in every dimension but in a rather trivial way: every polynomial automorphism is the composition of an affine map with a single root of a power series.…”

Time averages of polynomials

“…In [12] a generalization of locally finite maps was studied by Maubach and the author. Instead of looking at polynomial endomorphisms that are roots of polynomials as in Equation ( 3), one could consider the maps that are roots of power series:…”