J-stability in non-archimedean dynamics

Let T be a rational function of degree ≥ 2 on the Riemann sphere. Our results are based on the lemma that the diameter of a connected component of T−n(B(x, r)), centered at any point x in its Julia set J = J(T), does not exceed Lnrp for some constants L ≥ 1 and ρ > 0. Denote the transfer operator of a Hölder-continuous function φ on J satisfying P(T,φ) > supz∈Jφ(z). We study the behavior of {: n ≥ 1} for Hölder-continuous functions ψ and show that the sequence is (uniformly) normbounded in the space of Hölder-continuous functions for sufficiently small exponent. As a consequence we obtain that the density of the equilibrium measure μ for φ with respect to the -conformal measure is Höolder-continuous. We also prove that the rate of convergence of {ψ to this density in sup-norm is . From this we deduce the corresponding decay of the correlation integral and the central limit theorem for ψ.

We study a question on characterizing polynomials among rational functions of degree $$>1$$ on the projective line over an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value, from the viewpoint of dynamics and potential theory on the Berkovich projective line.

We show that Newton’s cubic methods (famous rational maps) have a locally connected Julia set except in some very specific cases. In particular, when those maps are infinitely renormalizable their Julia set is locally connected and contains small copies of nonlocally connected quadratic Julia sets. This also holds when Newton’s method is renormalizable and has Cremer points, unlike the polynomial case. After a dynamical description we show the necessity of the Brjuno condition within this family.

In this paper we consider singular perturbations of the quadratic polynomial $F(z) = z^2 + c$ where $c$ is the center of a hyperbolic component of the Mandelbrot set, i.e., rational maps of the form $z^2 + c + \lambda /z^2$. We show that, as $\lambda \rightarrow 0$, the Julia sets of these maps converge in the Hausdorff topology to the filled Julia set of the quadratic map $z^2 + c$. When $c$ lies in a hyperbolic component of the Mandelbrot set but not at its center, the situation is much simpler and the Julia sets do not converge to the filled Julia set of $z^2 + c$.

For Cantor circle Julia sets of hyperbolic rational maps, we prove that they are quasisymmetrically equivalent to standard Cantor circles (i.e., connected components are round circles). This gives a quasisymmetric uniformization of all Cantor circle Julia sets of hyperbolic rational maps. By analyzing the combinatorial information of the rational maps whose Julia sets are Cantor circles, we give a computational formula of the number of the Cantor circle hyperbolic components in the moduli space of rational maps for any fixed degree. We calculate the Hausdorff dimensions of the Julia sets which are Cantor circles, and prove that for any Cantor circle hyperbolic component <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the space of rational maps, the infimum of the Hausdorff dimensions of the Julia sets of the maps in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is equal to the conformal dimension of the Julia set of any representative <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f 0 element-of script upper H"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">f_0\in \mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and that the supremum of the Hausdorff dimensions is equal to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

This research aims to study complex dynamics, and one of the best examples of this concept is the Julia set, for this we will present new concepts specific to the Julia set of the rational map J(Gμ). We introduce the family of complex rational maps as: Gμ(z) = 2μm+1 z–m + z2m – μ3m+1 / zm (z2m – μm–1) , with m ≥ 2 and μ ϵ ℂ\{0}(ℂ complex numbers) such that μm+1 ≠ 1 and μ2m+2 ≠ 1. We prove J(Gμ) is a degenerate Sierpinski carpet or a Sierpinski carpet or a Cantor circle by condition one of the free critical points has an image for Gμ is diverge on ∞ or 0.

Let A be a rational function of degree at least 2 on the Riemann sphere. We say that A is tame if the algebraic curve A(x)-A(y)=0 has no factors of genus 0 or 1 distinct from the diagonal. In this paper, we show that if tame rational functions A and B have some orbits with infinite intersection, then A and B have a common iterate. We also show that for a tame rational function A decompositions of its iterates A^{\circ d}, d\geq 1, into compositions of rational functions can be obtained from decompositions of a single iterate A^{\circ N} for N large enough.

This paper is based on the author's previous work (Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products. Ergod. Th. & Dynam. Sys. 21 (2001), 563–603). We consider fiber-preserving complex dynamics on fiber bundles whose fibers are Riemann spheres and whose base spaces are compact metric spaces. In this context, without any assumption on (semi-)hyperbolicity, we show that the fiberwise Julia sets are uniformly perfect. From this result, we show that, for any semigroup G generated by a compact family of rational maps on the Riemann sphere ) that is semi-hyperbolic, except at perhaps finitely many points in the Julia set, and which satisfies the open set condition, is either porous or equal to the closure of the open set. Furthermore, we derive an upper estimate of the Hausdorff dimension of the Julia set.

Making use of the Hardy-Littlewood maximal function, we give a new proof of the following theorem of Pekarski: If /' is in L log L on a finite interval, then / can be approximated in the uniform norm by rational functions of degree n to an error 0(l/n) on that interval.

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In this article, we shall prove the following two basic results of arithmetic dynamics. Let ƒ: P¹(<i>K</i>) → P¹(<i>K</i>) be a rational function of degree 𝑑 &gt; 1. Discovered during the research, there are different results shown with the different definitions of <i>K</i>. If <i>K</i> is an algebraically closed field of characteristic 0, then <i>#Fix</i>(ƒ^<i>n</i>) = 𝑑^<i>n</i> + <i>0</i>(1). If <i>K</i> is a number field, then there are only finitely many preperiodic points of ƒ.

We study the quasisymmetric geometry of the Julia sets of McMullen maps $f_\lambda(z)=z^m+\lambda/z^\ell$, where $\ell$, $m\geq 2$ are integers satisfying $1/\ell+1/m<1$ and $\lambda\in\mathbb{C}\setminus\{0\}$. If the free critical points of $f_\lambda$ are escaped to the infinity, we prove that the Julia set $J_\lambda$ of $f_\lambda$ is quasisymmetrically equivalent to either a standard Cantor set, a standard Cantor set of circles or a round Sierpi\'{n}ski carpet (which is also standard in some sense). If the free critical points are not escaped, we give a sufficient condition on $\lambda$ such that $J_\lambda$ is a Sierpi\'{n}ski carpet and prove that most of them are quasisymmetrically equivalent to some round carpets. In particular, there exist infinitely renormalizable rational maps whose Julia sets are quasisymmetrically equivalent to round carpets.

Let be the Hardy space of functions that are analytic in the disk and let be the derivative of of order in the sense of Weyl. It is shown, for example, that if is a rational function of degree with all its poles in the domain , then , where , , and depends only on and .Bibliography: 32 titles.

Let be a Lebesgue measurable subset of a -dimensional cube (), let , where , and let be the least deviation of , in the metric of , from the rational functions of degree . If , then, for , has a local differential of order in the -metric at each point , except perhaps points of some set of metric dimension (this inequality is sharp). In addition, has a global differential of order in the metric of for any .Bibliography: 15 titles.

Denote by the least uniform deviation of the function , defined in a subset of -dimensional Euclidean space, from the rational functions of degree . It is shown that if , then, a.e. on , has a total differential. The case was previously treated by E. P. Dolženko. Bibliography: 9 titles.