84 research outputs found
Virtual Immediate Basins of Newton Maps and Asymptotic Values
Newton's root finding method applied to a (transcendental) entire function
f:C->C is the iteration of a meromorphic function N. It is well known that if
for some starting value z, Newton's method converges to a point x in C, then f
has a root at x. We show that in many cases, if an orbit converges to infinity
for Newton's method, then f has a `virtual root' at infinity. More precisely,
we show that if N has an invariant Baker domain that satisfies some mild
assumptions, then 0 is an asymptotic value for f.
Conversely, we show that if f has an asymptotic value of logarithmic type at
0, then the singularity over 0 is contained in an invariant Baker domain of N,
which we call a virtual immediate basin. We show by way of counterexamples that
this is not true for more general types of singularities.Comment: 15 pages, 1 figur
Scaling Ratios and Triangles in Siegel Disks
Let , where is a quadratic irrational.
McMullen proved that the Siegel disk for is self-similar about the critical
point. We give a lower bound for the ratio of self-similarity, and we show that
if is the golden mean, then there exists a triangle
contained in the Siegel disk, and with one vertex at the critical point. This
answers a 15 year old conjecture.Comment: 13 pages, 13 PostScript figure
From local to global analytic conjugacies
Let and be rational maps with Julia sets and , and let be any continuous map such that on . We show that if is -differentiable, with non-vanishing derivative, at some repelling periodic point , then admits an analytic extension to , where is the exceptional set of . Moreover, this extension is a semiconjugacy. This generalizes a result of Julia (Ann. Sci. École Norm. Sup. (3) 40 (1923), 97–150). Furthermore, if then the extended map is rational, and in this situation and , provided that is not constant. On the other hand, if then the extended map may be transcendental: for example, when is a power map (conjugate to ) or a Chebyshev map (conjugate to \pm \text{Х}_d with \text{Х}_d(z+z^{-1}) = z^d+z^{-d}), and when is an integral Lattès example (a quotient of the multiplication by an integer on a torus). Eremenko (Algebra i Analiz 1(4) (1989), 102–116) proved that these are the only such examples. We present a new proof
B\"ottcher coordinates
A well-known theorem of B\"ottcher asserts that an analytic germ
f:(C,0)->(C,0) which has a superattracting fixed point at 0, more precisely of
the form f(z) = az^k + o(z^k) for some a in C^*, is analytically conjugate to
z->az^k by an analytic germ phi:(C,0)->(C,0) which is tangent to the identity
at 0. In this article, we generalize this result to analytic maps of several
complex variables
Virtually repelling fixed points
In this article, we study the notion of virtually repelling fixed point. We first give a definition and an interpretation of it. We then prove that most proper holomorphic mappings f : U → V with U contained in V have at least one virtually repelling fixed point
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