bio  website  pcwww.liv.ac.uk/~lrempe 

location  Liverpool  
age  37  
visits  member for  5 years, 2 months 
seen  19 hours ago  
stats  profile views  963 
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 nearzeros 
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) finitedimensionality 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 EremenkoLyubich functions" also treats this Cauchy integral method in quite some generality. (arxiv.org/abs/1106.3439 , ams.org/mathscinetgetitem?mr=3214678 ). 
Dec 15 
comment 
GaussLucas 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 
GaussLucas type theorem for tracts and higher derivatives of a polynomial
@TrevorRichards Re: Question 2)  en.wikipedia.org/wiki/… 
Dec 12 
comment 
GaussLucas 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 
GaussLucas 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 higherorder 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 postcritically finite polynomials around repelling periodic points. 
Dec 11 
answered  GaussLucas 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 