2,195 reputation
818
bio website pcwww.liv.ac.uk/~lrempe
location Liverpool
age 37
visits member for 5 years, 2 months
seen 19 hours ago
Professor of Pure Mathematics at the University of Liverpool

2d
answered Extension of conformal map and annulus
2d
comment Extension of conformal map and annulus
Hi Neil, the answer is correct when the question is taken literally, i.e. "circle" actually means "round circle".
Apr
20
awarded  Nice Answer
Apr
13
answered Which way for reading the proofs?
Apr
7
answered Generalized Schwarz Lemma for near-zeros
Feb
16
answered Palis' conjecture and Newhouse's results
Jan
30
awarded  Yearling
Jan
26
comment What are the worst notations, in your opinion ?
Actually, I take issue with both of these notations. :) As a dynamicist, I would indeed agree that $\sin^2$ should be the second iterate of sine. However, $\sin$ is not invertible, and hence $\sin^{-1}$ should not be used for the arcsine, which is only a specific branch of the inverse function. $(\sin|_{[-\pi/2,\pi/2]})^{-1}$ would be ok I suppose ...
Jan
5
answered Complex function for mapping a circle to a superellipse
Dec
23
answered Is there any elementary proof of No wandering domain for polynomials
Dec
23
comment Is there any elementary proof of No wandering domain for polynomials
This is really still Sullivan's proof, however. In other words, it still uses quasiconformal deformations and (crucially) finite-dimensionality of the parameter space.
Dec
17
comment Entire function bounded at every line
My article "Hyperbolic entire functions with full hyperbolic dimension and approximation by Eremenko-Lyubich functions" also treats this Cauchy integral method in quite some generality. (arxiv.org/abs/1106.3439 , ams.org/mathscinet-getitem?mr=3214678 ).
Dec
15
comment Gauss--Lucas type theorem for tracts and higher derivatives of a polynomial
@TrevorRichards a) The convex hull of the set of zeros of an entire function will in general be much larger than the "tracts" in question, so $f$ would normally be expected to be unbounded on this convex hull (e.g. consider the zeros of the function e^{z^3}-1); b) There may be some limiting zeros of polynomials that "disappear", so that the convex hull of the zeros of the polynomials may not converge to the convex hull of the zeros of the limiting functions.
Dec
12
comment Gauss--Lucas type theorem for tracts and higher derivatives of a polynomial
@TrevorRichards Re: Question 2) - en.wikipedia.org/wiki/…
Dec
12
comment Gauss--Lucas type theorem for tracts and higher derivatives of a polynomial
@TrevorRichards Alternatively, if you are looking for explicit examples of polynomials, I suggest looking at some Shabat polynomials (= polynomial Belyi functions), i.e. polynomials with two critical values (or -1 and 1 may be the best normalisation for your question). Given any tree, with an embedding in the plane, there is a Shabat polynomial realising this tree. There are some programs for computing these (eg Don Marshall's "zipper", and Laurent Bartholdi also has a program). Not sure they're publicly available, but they exist. Just draw some "complicated" trees and experiment ..
Dec
12
comment Gauss--Lucas type theorem for tracts and higher derivatives of a polynomial
@TrevorRichards: The trouble with the "easy" examples is that they all essentially look like e^z, which does not have higher-order critical points ... One case of entire functions with a finite set of singular values that it's relatively easy to get your hands on, and that have quite different tracts from exponential maps, are Poincaré (linearising) functions of post-critically finite polynomials around repelling periodic points.
Dec
11
answered Gauss--Lucas type theorem for tracts and higher derivatives of a polynomial
Nov
25
revised A question on deficient values of entire functions
Corrected spelling and grammar.
Nov
25
suggested approved edit on A question on deficient values of entire functions
Nov
25
answered A question on deficient values of entire functions