数理解析研究所講究録 1447 巻 2005 年 90-107
90
Note on dynamically stable perturbations of parabolics Tomoki Kawahira
*
Nagoya University
[email protected]
Abstract In this note, we sketch some results on almost-dynamics-preserving perturbations of rational maps with parabolic cycles.
1
Introduction with rabbits
rabbit” has a friend called (“fat rabbit” at the root of Well known 1/3-limb of the Mandelbrot set. However the term “fat” does not sound good, so we tentatively call him “chubby rabbit”. “
$\mathrm{D}\mathrm{o}\mathrm{u}\mathrm{a}\mathrm{d}\mathrm{y}^{)}\mathrm{s}$
$|$
Figure 1: “plump”, “chubby” , and “overweight” . “Chubby rabbit” has a parabolic fixed point with 3 petals and multiplier . Actually there is an overweight rabbit in the main cardioid, which has an attracting fixed point with multiplier of the form $rc^{2\pi i/3}(00$
$1$
82
.
Notation.
. .
For a parabolic or attracting periodic point basin. $P(f)$
denotes the postcritical set of .
$C(f)$
denotes the critical set of .
$\alpha$
,
$A(\alpha)$
denotes its immediate
$f$
$f$
Polynomial case: Theorems of P.
2
$\mathrm{H}\dot{\mathrm{a}}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{k}\mathrm{y}$
In the case of polynomial, there are some results by Peter Haissinsky. Here we sketch his sequential work related to our question. In this section we assume that is a polynomial of degree $d\geq 2$ . $f$
2,1
Parabolic to repelling
The first theorem is on a perturbation of direction
$” \mathrm{p}$
arabolic
$arrow$
repelling”.
Theorem 2.1 (Haissinsky, [5]) If is geometrically finite with connected Julia accompanied by conjugaset, then there exists a polynomial perturbation are all subhype rbolic. cies between the actions of the Julia sets. Moreover, $f$
$f_{\epsilon}arrow f$
$f_{\epsilon}$
This theorem is extended later in \S 3. Sketch of the proof. The last sentence implies that every parabolic cycle in $J(f)$ is perturbed into a repelling cycle. We explicitly construct a rational perturbation $F_{\epsilon}=f+\epsilon R$ , where $R$ is a rational function which takes value zero at all parabolic cycles and at finite critical orbits on $J(f)$ . (Here we allow $\deg F_{\epsilon}\geq\deg f.)$ has cycles exactly the same places as the original Then parabolic cycles, but their multipliers are changed slightly by $R$ . Here we take a proper $R$ to make them repelling. Moreover, to preserve the local degree of critical orbits on the Julia sets, we take to have enough tangency at those points. If $\ll 1$ , we can take a nice topological-disk neighborhood of $J(f)$ such that is an analytic family of potynomia1-4ii map. By straightening, . Now it is known that the we obtain a subhyperbolic perturbation connected Julia sets of geometrically finite polynomials are locally connected. Thus every external ray land on thhe Julia sets. To check the dynamical stability on the Julia sets, we check the stability of the ray equivalence, which is defined by shared landing points of external rays. $F_{\epsilon}$
$R$
$U$
$\epsilon$
$\{U, F_{\epsilon}^{-1}(U), F_{\epsilon}\}$
$\mathrm{e}$
$f_{\epsilon}arrow f$
$\blacksquare$
83
Goldberg-Milnor conjecture. Theorem 2.1 gives an affirmative answer to the following Goldberg-M ilnor conjecture in the case of geometrically finite polynolnials: For a polynomial which has a parabolic cycle, there exists a small perturbation of such that $f$
.
.
$f$
the immediate basin of the parabolic cycle attracting cycles; and
the perturbed polynomial on its Julia set original polynomial on $J(f)$ .
$\iota s$
of some
converted to basins
is
topologically conjugate to the
$f$
Conversely, is it possible to create parabolics from hyperbolics(attracting or repelling)? The following results give us some partial answers.
Repelling to parabolic
2.2
Next we consider the opposite direction: “repelling theorem is: Theorem 2.2 a repelling fixed point (B)
$\beta$
on
is accessible from
$A(\alpha)$
Then there exists a polynomial such that
.
. @
$h|_{J(f)}$
is
Suppose has an attracting fixed point a and Vie also add the following condition:
and
for any a parabolic fixed point
gives
$\beta\not\in P(f)$
of
$g$
degree
$d$
and a homeomorphism
$z\in\hat{\mathbb{C}}-A(\alpha)$
$h\circ \mathrm{f}(\mathrm{z})=g\circ h(z)$
$h(\beta)$
.
$\partial A(\alpha)$
parabolic”. The second
$f$
$(\mathrm{H}\dot{\mathrm{a}}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{k}\mathrm{y}, [4])$
$\beta$
$arrow$
and
$h$
:
$\mathbb{C}$
$arrow \mathbb{C}$
;
$h(A(\alpha))=A(h(\beta))$
; and
a topological conjugacy between the actions on the Julia sets.
We can remove condition (B) when is geometrically finite. Moreover, we “ . can modify the statem ent by replacing the term “fixed point” with parabolic into the basin attracting an convert This theorem says that we can basin in our particular situation. The conjugacy breaks only on the immediate , where we operate tricky surgery by means of -conformal map. basin conforrnal map is not a quasiconformal map, though it is exponentially close to quasiconformal in some sense. be a measurable function which satisfies Let : $f$
$\mathrm{c}\mathrm{y}\mathrm{c}1\mathrm{e}^{)}’$
$A(\alpha)$
$\mathrm{n}\mathrm{s}$
$\mu$
$\mu-$
$\mathbb{C}arrow \mathrm{D}$
$\mu$
Area{z
$\in \mathbb{C}$
:
$|\mu(z)|>1-\in$
}
$\leq Ce^{-\eta/\epsilon}$
for some $C\geq 0$ and $\eta>0$ . Such a is called to be in the David class of functions can be 1 but it is quite close to the situation on C. Note that (that is, $\mu$
$||\mu||_{\infty}1-\epsilon$
} $=0$
84
), which will give us a quasiconformal map by solving the Beltram for a1J . equation Now the main tool is:
$\mathrm{i}$
$\epsilon
