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

location  Liverpool  
age  36  
visits  member for  4 years, 8 months 
seen  yesterday  
stats  profile views  875 
Professor of Pure Mathematics at the University of Liverpool
1d

awarded  Explainer 
Sep 19 
comment 
Continuous extension of Riemann maps and the CaratheodoryTorhorst Theorem
@MalikYounsi However, prime end theory is relevant in dynamics; for example, I think that it is implicitly used in PerezMarco's hedgehog theory to construct the analytic circle diffeomorphism associated to hedgehog dynamics. (One observes that the induced map on a onesided 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 nowandering domains theorem, though I may misremember. 
Sep 19 
comment 
Continuous extension of Riemann maps and the CaratheodoryTorhorst 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 CaratheodoryTorhorst Theorem
@AlexandreEremenko  Thanks! 
Sep 19 
revised 
Continuous extension of Riemann maps and the CaratheodoryTorhorst Theorem
corrected "postcomposition" to "precomposition", and added detail on the Torhorst Theorem. 
Sep 19 
asked  Continuous extension of Riemann maps and the CaratheodoryTorhorst 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 filledin Julia set in the case where this set has nonempty 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 