1,587 reputation
715
bio website pcwww.liv.ac.uk/~lrempe
location Liverpool
age 36
visits member for 4 years, 2 months
seen 3 hours ago
Professor of Pure Mathematics at the University of Liverpool

Apr
14
comment A question for hyperbolic metric in the proof for Bohr's lemma
I suspect that for 'on the boundary' you might want to read 'in the complement'.
Apr
13
comment 13 months and not even one report. what would you do?
Good to hear that there has finally been some progress. Hopefully this whole experience will be over for you soon.
Apr
13
answered A question for hyperbolic metric in the proof for Bohr's lemma
Apr
13
awarded  Revival
Feb
22
awarded  Enlightened
Feb
22
awarded  Nice Answer
Feb
8
answered Extending a function from $\mathbb{Q}$ to the upper half plane $\mathbb{H}\cup\mathbb{Q}\cup\{i\infty\}$
Jan
30
awarded  Yearling
Dec
11
accepted Belyi functions on non-compact surfaces; or: Building Riemann surfaces from equilateral triangles
Nov
19
comment Entire functions with a null real escaping set
The answer is positive if the function belongs to the class $\mathcal{B}$ where the set of singular values is bounded. (Here the real escaping set, if nonempty, must contain an interval of the form $[R,\infty)$ or $(-\infty,R]$.)
Nov
19
answered Dynamical properties of injective continuous functions on $\mathbb{R}^d$
Nov
19
comment Contractibility of connected holomorphic dynamics?
If by "holomorphic function" you mean "entire function", then the answer is yes (but this is really a different question). It may not be quite as trivial for transcendental maps as for polynomials, but is still true.
Nov
18
comment Contractibility of connected holomorphic dynamics?
Yes, they always have trivial fundamental group. This is trivial using standard results. After giving some thought to the matter, I believe, however, that Cremer quadratic Julia sets are quite unlikely to be path-connected.
Nov
18
comment Contractibility of connected holomorphic dynamics?
@Sebastien A Cremer Julia set has no interior and can never disconnect the plane, so - if I have my definitions right - every such Julia set is 1-connected.
Nov
18
revised Contractibility of connected holomorphic dynamics?
added 768 characters in body
Nov
18
comment Contractibility of connected holomorphic dynamics?
@Alex - I will clarify. Sebastien - I don't understand. I believe contractibility is a stronger property than being 1-connected. In any case, I am not sure that it is known whether Cremer Julia sets can be pathwise connected.
Nov
17
answered Contractibility of connected holomorphic dynamics?
Nov
17
comment Contractibility of connected holomorphic dynamics?
For entire functions, the question is even more weird, because the set is no longer closed (nor open). Distinctions such as whether the set itself is connected, or only connected when you add infinity, will also become important.
Nov
17
comment Contractibility of connected holomorphic dynamics?
I believe that the answer is negative, probably for every quadratic Cremer polynomial. It shouldn't be too hard to prove, but I probably won't have a chance to think about it carefully enough for a couple of days. If no-one has written an answer by then, I will try.
Nov
15
comment Belyi functions on non-compact surfaces; or: Building Riemann surfaces from equilateral triangles
But the point is that - using methods developed by Chris Bishop - we can make sure that the surface is quite close to the original one - close enough to ensure that we can actually compensate for it in the next step, and in the limit obtain a function actually defined on the desired surface.