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

1d
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 ...
Aug
29
answered Algorithm for determining when polynomial iteration is bounded?
Aug
10
comment When is a Newton basin fractal continuously determined by the roots of its polynomial?
Even in the Mandelbrot set, it is not known whether the only stable parameters are given by those where the critical point tends to attracting cycles. This ("Density of Hyperbolicity") is a very famous conjecture, and would follow from local connectivity of the Mandelbrot set (which is perhaps the most famous open problem in the entire field).
Aug
9
answered When is a Newton basin fractal continuously determined by the roots of its polynomial?
Jul
29
answered Examples of cubic Julia sets
Jul
2
awarded  Curious
Jun
23
answered Is there a effective computational criterion to all periodic points of a rational function are repelling.
Jun
9
answered existence of rational functions with prescribed critical values and ramification degrees at critical points