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bio website pcwww.liv.ac.uk/~lrempe
location Liverpool
age 36
visits member for 4 years, 8 months
seen 12 hours ago
Professor of Pure Mathematics at the University of Liverpool

Oct
16
comment Conditions conformal mapping to be expansive
@ChrisJudge, I have elaborated further. Hope this makes sense.
Oct
16
revised Conditions conformal mapping to be expansive
added further detail and ideas to the answer
Oct
16
answered Conditions conformal mapping to be expansive
Oct
3
comment Metric properties of a quadratic differential at an essential singularity
@XinNie - Exactly!
Oct
3
comment Metric properties of a quadratic differential at an essential singularity
I am well aware of Picard's theorem. Still, you have not justified how the theorem should imply an answer to Problem 1. The function will have to grow very fast towards the singularity somewhere, but the set where it does so may be very small in measure. Hence you should need a more quantitative argument.
Oct
2
comment Metric properties of a quadratic differential at an essential singularity
How would you get Problem 1 just from Picard's theorem? Surely you need something more quantitative, since the part of the punctured disc where $f$ is large could be very small.
Oct
2
comment Metric properties of a quadratic differential at an essential singularity
How do you conclude your inequality? After all, there may well be some places where $f$ is much smaller than the corresponding term in the Laurent series, so it seems that you need to give some justification.
Sep
30
awarded  Explainer
Sep
19
comment Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem
@MalikYounsi However, prime end theory is relevant in dynamics; for example, I think that it is implicitly used in Perez-Marco's hedgehog theory to construct the analytic circle diffeomorphism associated to hedgehog dynamics. (One observes that the induced map on a one-sided neighbourhood of the unit circle extends continuously, and then applies Schwarz reflection.) Also, I think it was used in Sullivan's original proof of the no-wandering domains theorem, though I may misremember.
Sep
19
comment Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem
@MalikYounsi - It is rather the case that the continuous extension of the Riemann map is the real reason we are interested in MLC (as it would give us a complete topological model of the Mandelbrot set, as well as combinatorial rigidity and density of hyperbolicity).
Sep
19
comment Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem
@AlexandreEremenko - Thanks!
Sep
19
revised Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem
corrected "post-composition" to "pre-composition", and added detail on the Torhorst Theorem.
Sep
19
asked Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem
Sep
19
revised Integrability of second derivative of conformal mappings
added link
Sep
19
comment Integrability of second derivative of conformal mappings
Another place to look for results and references may Pommerenke's "Boundary Behaviour of conformal maps". I don't have it to hand at the moment.
Sep
19
comment Integrability of second derivative of conformal mappings
Glancing over the paper, the final part of their Theorem states that there exist domains with smooth boundary so that the derivative of the mapping function does not have Bounded Mean Oscillation. This does not seem to formally imply an answer to the question (?), but may be a good place to start.
Sep
19
suggested suggested edit on Integrability of second derivative of conformal mappings
Sep
8
comment Periodicities of a Complex Dynamical System
Essentially the same question posted previously by the same user at mathoverflow.net/questions/180278/…
Sep
8
comment Boundedness and Convergence of a Complex sequence
Duplicate of mathoverflow.net/questions/180331/… , which has also been posted math.stackexchange math.stackexchange.com/questions/923353/…
Aug
29
comment Algorithm for determining when polynomial iteration is bounded?
@Per 'in the filled version, you can essentially paint all points that converges nicely (which are outside the julia set), so this is no surprise. The points in the julia set are the "hard" ones.' - On the contrary, it is harder to prove computability of the filled-in Julia set in the case where this set has non-empty interior ...