Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Simple Proofs for the Derivative Estimates of the Holomorphic Motion
  near Two Boundary Points of the Mandelbrot Set   For the complex quadratic family $q_c:z\mapsto z^2+c$, it is known that every
point in the Julia set $J(q_c)$ moves holomorphically on $c$ except at the
boundary points of the Mandelbrot set. In this note, we present short proofs of
the following derivative estimates of the motions near the boundary points
$1/4$ and $-2$: for each $z = z(c)$ in the Julia set, the derivative $dz(c)/dc$
is uniformly $O(1/\sqrt{1/4-c})$ when real $c\nearrow 1/4$; and is uniformly
$O(1/\sqrt{-2-c})$ when real $c\nearrow -2$. These estimates of the derivative
imply Hausdorff convergence of the Julia set $J(q_c)$ when $c$ approaches these
boundary points. In particular, the Hausdorff distance between $J(q_c)$ with
$0\le c<1/4$ and $J(q_{1/4})$ is exactly $\sqrt{1/4-c}$.

11
The categories of this paper are: hep-lat
31
The categories of this paper are: ao-sci
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex
20
The categories of this paper are: acc-phys
29
The categories of this paper are: funct-an
7
The categories of this paper are: hep-th
19
The categories of this paper are: eess
23
The categories of this paper are: solv-int
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The categories of this paper are: dg-ga
30
The categories of this paper are: q-alg
33
The categories of this paper are: plasm-ph
1
The categories of this paper are: cs
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The categories of this paper are: cmp-lg
10
The categories of this paper are: q-bio
32
The categories of this paper are: atom-ph