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bio website pcwww.liv.ac.uk/~lrempe
location Liverpool
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Professor of Pure Mathematics at the University of Liverpool

1d
answered A question on $J(f)$ and $J(f')$
Jul
16
comment Growth of the size of iterated polynomials
I am a little confused - but may have missed some subtlety. 1) Should $\log$ not be $\log_+$ (i.e., maximum of 0 and the logarithm)? Otherwise, what happens for $p(z)=z^2$ and $a=0$? 2) If this is the case, why isn't the convergence a consequence of [the proof of] Boettcher's theorem?
Jul
13
revised Do the bounded isophase lines of a complex polynomial $f$ through the zeroes of $f’$ define a spanning tree?
added 1037 characters in body
Jul
13
comment Do the bounded isophase lines of a complex polynomial $f$ through the zeroes of $f’$ define a spanning tree?
Ah, this is essentially the same as Alex's corrected answer (which was not there when I started typing mine). I will leave it because maybe there are some additional details that might be helpful.
Jul
13
answered Do the bounded isophase lines of a complex polynomial $f$ through the zeroes of $f’$ define a spanning tree?
Jul
1
comment Is Every Holomorphic Near an Entire?
In addition to the answers given, a literature reference: Dieter Gaier, "Lectures on Complex Approximation", includes Mergelyan's and Arakelyan's theorems, and much more!
Jun
12
answered Equivalence of Definitions of Quasiconformal Surfaces?
Jun
8
revised Can the image of a disk have nontrivial Hausdorff measure for $1 < d < 2$?
added 75 characters in body
Jun
8
answered Can the image of a disk have nontrivial Hausdorff measure for $1 < d < 2$?
Jun
8
comment Anti-Mandelbrot set
Higher-degree (unicritical) versions are sometimes called "multicorns". I believe Dierk Schleicher is among the people who has done some work on the topic (after Milnor).
Jun
5
comment Fixed point property for intersection of spaces which are homeomorphic to a disk
I think you mean "does not separate the plane" in the Question.
Jun
2
comment Is this a rational function?
Any globally defined meromorphic function that has a removable singularity (as a function on the Riemann sphere) at $\infty$ is clearly rational. Hence any non-rational meromorphic function in the plane has an essential singularity at infinity, and is transcendental.
Jun
1
comment Is this a rational function?
@GeraldEdgar - surely this follows also from the first argument given in this answer? Indeed, any globally defined meromorphic function is either rational or transcendental.
May
26
awarded  Nice Question
May
24
revised Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem
clarified what is and is not statedin Torhorst's paper. Also added link for Wilder's paper.
May
23
comment Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem
Not sure if Torhorst explicitly mentions continuous extension but this is clearly understood. Will try to clarify in question when I get the chance.
May
23
comment Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem
Thanks for the interest. Posting from mobile so apologies for typos. 1. Carathéodory considered SIMPLE closed curves, not general curves. 2. Carathéodory proves continuous extension iff all prime ends of first kind. Torhorst proves and states all prime ends of first kind iff lc.
May
22
awarded  Promoter
May
22
answered Non-bijective conformal maps between annuli
May
22
revised Conformal map and Jordan curve
Provided further details for the argument and adjusted notation.