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Han Peters

Publications and source records attributed to Han Peters.

43 records · Page 3Linked to original sources
arXiv2007

Time averages of polynomials

We define and study when a polynomial mapping has a local or global time average. We conjecture that a polynomial f in the complex plane has a time average near a point z if and only if z is eventually mapped into a Siegel-disc of f. We prove that the conjecture holds generically, namely for those polynomials whose iterates have the maximal number of critical values. Important steps in the proofs rely on understanding the iterated monodromy groups. We also show that a polynomial automorphism of C^2 has a global time average if and only if the map is conjugate to an elementary mapping. The definition of a time average is motivated by an attempt to understand the polynomial automorphism groups in dimensions 3 and higher.

math.CV↗
arXiv2007

Polynomial maps that are roots of power series

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.

math.CV↗
arXiv2007

Degree Estimates for Polynomials Constant on a Hyperplane

The study of proper rational mappings between balls in complex Euclidean spaces naturally leads to the relationship between the degree and imbedding dimension of such a mapping. The special case for monomial mappings is equivalent to the question discussed in this paper. Estimate the degree $d$ of a polynomial in $n$ real variables, assumed to have non-negative coefficients and to be constant on a hyperplane, in terms of the number $N$ of its terms. No such estimate is possible when $n=1$. The sharp bound $d\le 2N-3$ is known when $n=2$. This paper includes two main results. The first provides a bound, not sharp for $n\ge 3$, for all $n\ge 2$. This bound implies the more easily stated bound $d\le {4(2N-3)\over 3(2n-3)}$ for $n\ge 3$. The second result is a stabilization theorem; if $n$ is sufficiently large given $d$, then the sharp bound $d \le {N-1 \over n-1}$ holds. In this situation we determine all polynomials for which the bound is sharp.

math.CV↗
arXiv2006

Attracting basins of volume preserving automorphisms of $\mathbb{C}^k$

We study toplogical properties of attracting sets for automorphisms of $\mathbb{C}^k$. Our main result is that a generic volume preserving automorphism has a hyperbolic fixed point with a dense stable manifold. We prove the same result for volume preserving maps tangent to the identity. On the other hand, we show that an attracting set can only contain a neighborhood of the fixed point if the fixed point is an attracting fixed point. We will see that the latter does not hold in the non-autonomous setting.

math.CV↗
arXiv2004

Perturbed Basins of Attraction

Let F be an automorphism of C^k which has a fixed point. It is well known that the basin of attraction is biholomorphically equivalent to C^k. We will show that the basin of attraction of a sequence of automorphisms is also biholomorphic to C^k if all the automorphisms are small perturbations of the original map F.

math.CV↗
arXiv2004

Non-Autonomous Basins of Attraction With 4-Dimensional Boundaries

We study whether the basin of attraction of a sequence of automorphisms of $\mathbb{C}^k$ is biholomorphic to $\mathbb{C}^k$. In particular we show that given any sequence of automorphisms with the same attracting fixed point, the basin is biholomorphic to $\mathbb{C}^k$ if the maps are repeated often enough. We also construct Fatou-Bieberbach domains whose boundaries are 4-dimensional.

math.CV↗