اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدConnectedness properties of the set where the iterates of an entire function are bounded
111
0
0.0
(
0
)
تأليف
John Osborne
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We investigate some connectedness properties of the set of points K(f) where the iterates of an entire function f are bounded. In particular, we describe a class of transcendental entire functions for which an analogue of the Branner-Hubbard conjecture holds and show that, for such functions, if K(f) is disconnected then it has uncountably many components. We give examples to show that K(f) can be totally disconnected, and we use quasiconformal surgery to construct a function for which K(f) has a component with empty interior that is not a singleton.
قيم البحث
اقرأ أيضاً
For a rational function f we consider the norm of the derivative with respect to the spherical metric and denote by K(f) the supremum of this norm. We give estimates of this quantity K(f) both for an individual function and for sequences of iterates.
We prove that along any marked point the Green function of a meromorphic family of polynomials parameterized by the punctured unit disk explodes exponentially fast near the origin with a continuous error term.
The multiple Birkhoff recurrence theorem states that for any $dinmathbb N$, every system $(X,T)$ has a multiply recurrent point $x$, i.e. $(x,x,ldots, x)$ is recurrent under $tau_d=:Ttimes T^2times ldots times T^d$. It is natural to ask if there alwa
ys is a multiply minimal point, i.e. a point $x$ such that $(x,x,ldots,x)$ is $tau_d$-minimal. A negative answer is presented in this paper via studying the horocycle flows. However, it is shown that for any minimal system $(X,T)$ and any non-empty open set $U$, there is $xin U$ such that ${nin{mathbb Z}: T^nxin U, ldots, T^{dn}xin U}$ is piecewise syndetic; and that for a PI minimal system, any $M$-subsystem of $(X^d, tau_d)$ is minimal.
We prove that there exists a homeomorphism $chi$ between the connectedness locus $mathcal{M}_{Gamma}$ for the family $mathcal{F}_a$ of $(2:2)$ holomorphic correspondences introduced by Bullett and Penrose, and the parabolic Mandelbrot set $mathcal{M}
_1$. The homeomorphism $chi$ is dynamical ($mathcal{F}_a$ is a mating between $PSL(2,mathbb{Z})$ and $P_{chi(a)}$), it is conformal on the interior of $mathcal{M}_{Gamma}$, and it extends to a homeomorphism between suitably defined neighbourhoods in the respective one parameter moduli spaces. Following the recent proof by Petersen and Roesch that $mathcal{M}_1$ is homeomorphic to the classical Mandelbrot set $mathcal{M}$, we deduce that $mathcal{M}_{Gamma}$ is homeomorphic to $mathcal{M}$.
We establish the Geometric Dynamical Northcott Property for polarized endomorphisms of a projective normal variety over a function field $mathbf{K}$ of characteristic zero. This extends previous results of Benedetto, Baker and DeMarco in dimension $1
$, and of Chatzidakis-Hrushovski in higher dimension. Our proof uses complex dynamics arguments and does not rely on the previous one. We first show that when $mathbf{K}$ is the field of rational functions of a smooth complex projective variety, the canonical height of a subvariety is the mass of the appropriate bifurcation current and that a marked point is stable if and only if its canonical height is zero. We then establish the Geometric Dynamical Northcott Property using a similarity argument. Moving from points to subvarieties, we propose, for polarized endomorphisms, a dynamical version of the Geometric Bogomolov Conjecture, recently proved by Cantat, Gao, Habegger and Xie. We establish several cases of this conjecture notably non-isotrivial polynomial skew-product with an isotrivial first coordinate.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
