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bio website pcwww.liv.ac.uk/~lrempe
location Liverpool
age 36
visits member for 4 years, 10 months
seen 2 days ago
Professor of Pure Mathematics at the University of Liverpool

Dec
17
comment Entire function bounded at every line
My article "Hyperbolic entire functions with full hyperbolic dimension and approximation by Eremenko-Lyubich functions" also treats this Cauchy integral method in quite some generality. (arxiv.org/abs/1106.3439 , ams.org/mathscinet-getitem?mr=3214678 ).
Dec
15
comment Gauss--Lucas type theorem for tracts and higher derivatives of a polynomial
@TrevorRichards a) The convex hull of the set of zeros of an entire function will in general be much larger than the "tracts" in question, so $f$ would normally be expected to be unbounded on this convex hull (e.g. consider the zeros of the function e^{z^3}-1); b) There may be some limiting zeros of polynomials that "disappear", so that the convex hull of the zeros of the polynomials may not converge to the convex hull of the zeros of the limiting functions.
Dec
12
comment Gauss--Lucas type theorem for tracts and higher derivatives of a polynomial
@TrevorRichards Re: Question 2) - en.wikipedia.org/wiki/…
Dec
12
comment Gauss--Lucas type theorem for tracts and higher derivatives of a polynomial
@TrevorRichards Alternatively, if you are looking for explicit examples of polynomials, I suggest looking at some Shabat polynomials (= polynomial Belyi functions), i.e. polynomials with two critical values (or -1 and 1 may be the best normalisation for your question). Given any tree, with an embedding in the plane, there is a Shabat polynomial realising this tree. There are some programs for computing these (eg Don Marshall's "zipper", and Laurent Bartholdi also has a program). Not sure they're publicly available, but they exist. Just draw some "complicated" trees and experiment ..
Dec
12
comment Gauss--Lucas type theorem for tracts and higher derivatives of a polynomial
@TrevorRichards: The trouble with the "easy" examples is that they all essentially look like e^z, which does not have higher-order critical points ... One case of entire functions with a finite set of singular values that it's relatively easy to get your hands on, and that have quite different tracts from exponential maps, are Poincaré (linearising) functions of post-critically finite polynomials around repelling periodic points.
Dec
11
answered Gauss--Lucas type theorem for tracts and higher derivatives of a polynomial
Nov
25
revised A question on deficient values of entire functions
Corrected spelling and grammar.
Nov
25
suggested approved edit on A question on deficient values of entire functions
Nov
25
answered A question on deficient values of entire functions
Nov
25
revised Is there an (almost) dense set of quadratic polynomials which is not in the interior of the Mandelbrot set?
Extended the answer.
Nov
24
revised Is there an (almost) dense set of quadratic polynomials which is not in the interior of the Mandelbrot set?
Slightly reformulated the question to make it more descriptive. Fixed some grammar/wording, and tried to add some clarification (hopefully without changing the intention of the author or basic structure of the question).
Nov
24
suggested approved edit on Is there an (almost) dense set of quadratic polynomials which is not in the interior of the Mandelbrot set?
Nov
24
answered Is there an (almost) dense set of quadratic polynomials which is not in the interior of the Mandelbrot set?
Nov
2
comment 13 months and not even one report. what would you do?
Know it's a bit late, but glad to hear that things worked out.
Nov
2
answered Failure of Mostow rigidity in dim. 2
Oct
31
answered Is it possible to define the density of the logistic map for $x<0$?
Oct
27
comment Clustering of periodic points for a polynomial iteration of $\mathbb{C}$
@VesselinDimitrov Yes, that's what I took your first comment to mean. Interestingly, using recent work on 'near-parabolic renormalization', it could be possible to quantify the above construction, and hence - combined with your observation - give estimates on the combinatorics (i.e., the numbers $q_j$) that ensure that the resulting parameter is transcendental. That could be a publishable result, although depending on the technical work needed it may or may not be worthwhile to work it out. Davoud Cheraghi (Imperial College) is a world expert on this type of argument.
Oct
26
comment Clustering of periodic points for a polynomial iteration of $\mathbb{C}$
@VesselinDimitrov that's an interesting point. The construction is a generalization of the Feigenbaum point (where we make a sequence of period 2 bifurcations, and take the limit). In this case, I believe that it is not known whether the limit parameter is transcendental. (More generally, one could ask whether every infinitely renormalizable parameter is transcendental.)
Oct
26
comment A forked plane continuum
Nice question. I suspect that the answer is no, i.e. a continuum may exist that violates your requirement. I would try to construct it as a nested intersection of continua $K_n$, where every subcontinuum of $K_n$ that contains both $P$ and a point in either of the two quadrants at height $q_n$ must also contain a point at the same height in the other quadrant. Here $q_n$ is a sequence that is dense in the interval $[0,1]$. The construction in each step would be based on a thickened version of the $\sin(1/x)$-continuum. I cannot claim to have thought out the details, so this may not work ...
Oct
26
answered Clustering of periodic points for a polynomial iteration of $\mathbb{C}$