1,726 reputation
716
bio website pcwww.liv.ac.uk/~lrempe
location Liverpool
age 36
visits member for 4 years, 7 months
seen 8 hours ago
Professor of Pure Mathematics at the University of Liverpool

14h
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.
14h
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).
16h
comment Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem
@AlexandreEremenko - Thanks!
16h
revised Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem
corrected "post-composition" to "pre-composition", and added detail on the Torhorst Theorem.
17h
asked Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem
18h
revised Integrability of second derivative of conformal mappings
added link
19h
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.
19h
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.
19h
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
May
25
comment Original article about a theorem of Cartan on iterations of analytic functions
If you are looking for a generalization in complex dimension 1, then this is given by the (hyperbolic version of) the Schwarz lemma.