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We answer a question of Schleicher by showing that, for an exponential map with nonescaping singular value, every periodic ray lands. This is an analog of a theorem of Douady and Hubbard concerning polynomials. We also prove a partial converse: there are periodic external rays landing at all periodic points, with the exception of at most one periodic orbit.
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In this article, we present a landing theorem for periodic dynamic rays for transcendental entire maps which have bounded post-singular sets, by using standard hyperbolic geometry results.
Based on the distortion theory developed by Cui--Tan \cite{CT15}, we prove the landing of every parameter ray at critical portraits coming from non-recurrent polynomials, thereby generalizing a result of Kiwi \cite[Corollary]{Ki05}
We give a criterion to determine when two external rays land at the same point for polynomials with locally connected Julia sets. As an application, we provide an elementary proof of the monotonicity of the core entropy along arbitrary veins of the Mandelbrot set.
We show that repelling periodic points are landing points of periodic rays for exponential maps whose singular value has bounded orbit. For polynomials with connected Julia sets, this is a celebrated theorem by Douady, for which we present a new proof. In both cases we also show that points in hyperbolic sets are accessible by at least one and at most finitely many rays. For exponentials this allows us to conclude that the singular value itself is accessible.
It is well known that every rational parameter ray of the Mandelbrot set lands at a single parameter. We study the rational parameter rays of the multicorn $\mathcal{M}_d^*$, the connectedness locus of unicritical antiholomorphic polynomials of degree $d$, and give a complete description of their accumulation properties. One of the principal results is that the parameter rays accumulating on the boundaries of odd period (except period $1$) hyperbolic components of the multicorns do not land, but accumulate on arcs of positive length consisting of parabolic parameters. We also show the existenc…
A transcendental entire function f is called geometrically finite if the intersection of the set of singular values with the Fatou set is compact and the intersection of the postsingular set with the Julia set is finite. (In particular, this includes all entire functions with finite postsingular set.) If f is geometrically finite, then the Fatou set of f is either empty or consists of the basins of attraction of finitely many attracting or parabolic cycles. Let z_0 be a repelling or parabolic periodic point of such a map f. We show that, if f has finite order, then there exists an injective cu…
We generalize a combinatorial formula of Douady from the main cardioid to other hyperbolic components $H$ of the Mandelbrot set, constructing an explicit piecewise linear map which sends the set of angles of external rays landing on $H$ to the set of angles of external rays landing on the real axis.
We prove a structure theorem for the multibrot sets, which are the higher degree analogues of the Mandelbrot set, and give a complete picture of the landing behavior of the rational parameter rays and the bifurcation phenomenon. Our proof is inspired by previous works of Schleicher and Milnor on the combinatorics of the Mandelbrot set; in particular, we make essential use of combinatorial tools such as orbit portraits and kneading sequences. However, we avoid the standard global counting arguments in our proof and replace them by local analytic arguments to show that the parabolic and the Misi…
Periodic Points and Smooth Rays
本文证明对于具有不连通填充 Julia 集的多项式，其非退化连通分量中的斥性或抛物周期点必为至少一条光滑外部射线的着陆点，且该结论最优。
In this paper, we use the Carathéodory Convergence Theory to prove a landing theorem of rays in hyperbolic components with rational arguments. Although the proof is done in the setting of a family of entire transcendental maps with two singular values, the method can be generalized to many other one parameter families, with some modifications.
In this article, we prove that for several one-dimensional holomorphic families of holomorphic maps, in the parameter plane, there exists a local piece of a curve that lands at a given parabolic parameter, in the spirit of well-known results about the quadratic and the exponential families. We also show that, under some assumptions, this general result partially answers the existence and landing questions of ray structures in the parameter planes for holomorphic families of transcendental entire maps.
We investigate the set $I$ of parameters $κ$ for which the singular value of $z\mapsto e^z+κ$ converges to $\infty$. The set $I$ consists of uncountably many parameter rays, plus landing points of some of these rays. We show that the parameter rays have Hausdorff dimension 1, while the ray endpoints in $I$ alone have dimension 2. Analogous results were known for dynamical planes of exponential maps; our result shows that this also holds in parameter space.
We study the distribution of periodic points for a wide class of maps, namely entire transcendental functions of finite order and with bounded set of singular values, or compositions thereof. Fix $p\in\N$ and assume that all dynamic rays which are invariant under $f^p$ land. An interior $p$-periodic point is a fixed point of $f^p$ which is not the landing point of any periodic ray invariant under $f^p$. Points belonging to attracting, Siegel or Cremer cycles are examples of interior periodic points. For functions as above we show that rays which are invariant under $f^p$, together with their l…
We discuss in detail the dynamics of maps $z\mapsto ae^z+be^{-z}$ for which both critical orbits are strictly preperiodic. The points which converge to $\infty$ under iteration contain a set $R$ consisting of uncountably many curves called ``rays'', each connecting $\infty$ to a well-defined ``landing point'' in $\C$, so that every point in $\C$ is either on a unique ray or the landing point of finitely many rays. The key features of this paper are the following two: (1) this is the first example of a transcendental dynamical system where the Julia set is all of $\C$ and the dynamics is descri…
We consider growth conditions for (frequently) Birkhoff-universal functions of exponential type with respect to the different rays emanating from the origin. For that purpose, we investigate their (conjugate) indicator diagram or, equivalently, their indicator function. Some known results, where growth is measured with respect to the maximum modulus, are extended.
A landing theorem for entire functions with bounded post-singular sets
将多项式动力学的Douady-Hubbard着陆定理推广至具有有界后奇异集的整函数，证明周期毛或周期丝状物与斥型或抛物型周期点互为着陆点。
We consider the case of an exponential map for which the singular value is accessible from the set of escaping points. We show that there are dynamic rays of which do not land. In particular, there is no analog of Douady's ``pinched disk model'' for exponential maps whose singular value belongs to the Julia set. We also prove that the boundary of a Siegel disk $U$ for which the singular value is accessible both from the set of escaping points and from $U$ contains uncountably many indecomposable continua.
Let f be an entire function whose set of singular values is bounded and suppose that f has a Siegel disk such that f restricts to a homeomorphism of the boundary. We show that the Siegel disk is bounded. Using a result of Herman, we deduce that if additionally the rotation number of the Siegel disk is Diophantine, then its boundary contains a critical point of f. Suppose furthermore that all singular values of f lie in the Julia set. We prove that, if f has a Siegel disk $U$ whose boundary contains no singular values, then the condition that f is a homeomorphism of the boundary of U is automat…
We investigate the set of parameters $κ\in\C$ for which the singular orbit $(0,e^κ,...)$ of $E_κ(z):=\exp(z+κ)$ converges to $\infty$. These parameters are organized in smooth curves in parameter space called \emph{parameter rays}.
We study the bifurcation loci of quadratic (and unicritical) polynomials and exponential maps. We outline a proof that the exponential bifurcation locus is connected; this is an analog to Douady and Hubbard's celebrated theorem that (the boundary of) the Mandelbrot set is connected. For these parameter spaces, a fundamental conjecture is that hyperbolic dynamics is dense. For quadratic polynomials, this would follow from the famous stronger conjecture that the bifurcation locus (or equivalently the Mandelbrot set) is locally connected. It turns out that a formally slightly weaker statement is …
A key point in Douady and Hubbard's study of the Mandelbrot set $M$ is the theorem that every parabolic point $c\ne 1/4$ in $M$ is the landing point for exactly two external rays with angle which are periodic under doubling. This note will try to provide a proof of this result and some of its consequences which relies as much as possible on elementary combinatorics, rather than on more difficult analysis. It was inspired by section 2 of the recent thesis of Schleicher (see also Stony Brook IMS preprint 1994/19, with E. Lau), which contains very substantial simplifications of the Douady-Hubbard…
We consider entire transcendental maps with bounded set of singular values such that periodic rays exist and land. For such maps, we prove a refined version of the Fatou-Shishikura inequality which takes into account rationally invisible periodic orbits, that is, repelling cycles which are not landing points of any periodic ray. More precisely, if there are $q<\infty$ singular orbits, then the sum of the number of attracting, parabolic, Siegel, Cremer or rationally invisible orbits is bounded above by $q$. In particular, there are at most $q$ rationally invisible repelling periodic orbits. The…
Cubic Polynomial Maps with Periodic Critical Orbit, Part III: Tessellations and Orbit Portraits
本文研究三次多项式映射参数空间中周期临界轨道的动力学变化，通过逃逸区域、参数射线落点及周期q镶嵌刻画参数射线与动态射线的关系，并证明落点必为抛物映射或Misiurewicz映射。
Dynamical approximation of postsingularly finite exponentials
本文构造了一列后临界有限单临界多项式局部一致逼近任意后奇异有限指数函数，并揭示了参数射线角度与指数参数平面外部地址之间的关系。
We consider transcendental entire functions that are compositions of a polynomial and the exponential for which all singular values escape on disjoint rays. Based on their classification in [B3] we investigate their dependence on parameters, that is, on the potentials and external addresses of singular values. Using continuity argument we generalize the classification in [B3] to the case when dynamic rays containing singular values are (pre-)periodic.
We develop an abstract model for the dynamics of an exponential map $z\mapsto \exp(z)+κ$ on its set of escaping points and, as an analog of Boettcher's theorem for polynomials, show that every exponential map is conjugate, on a suitable subset of its set of escaping points, to a restriction of this model dynamics. Furthermore, we show that any two attracting and parabolic exponential maps are conjugate on their sets of escaping points; in fact, we construct an analog of Douady's "pinched disk model" for the Julia sets of these maps. On the other hand, we show that two exponential maps are gene…
Orbit portraits were introduced by Milnor as a combinatorial tool to describe the patterns of all periodic dynamical rays landing on a periodic cycle of a quadratic polynomial. This encodes information about the dynamics and the parameter spaces of these maps. We carry out a similar analysis for unicritical anti-polynomials and give an explicit description of the orbit portraits that can occur for such maps in terms of their characteristic angles, which turns out to be rather restricted when compared with the holomorphic case. Finally, we prove a realization theorem for these combinatorial obj…
We show that an exponential map $f_c(z)=e^z+c$ whose singular value $c$ is combinatorially non-recurrent and non-escaping is uniquely determined by its combinatorics, i.e. the pattern in which its dynamic rays land together. We do this by constructing puzzles and parapuzzles in the exponential family. We also prove a theorem about hyperbolicity of the postsingular set in the case that the singular value is non-recurrent. Finally, we show that boundedness of the postsingular set implies combinatorial non-recurrence if $c$ is in the Julia set.
Length spectrum of periodic rays for billard flow
本文研究了周期射线长度谱的性质，证明了在弱分离条件下，周期轨道的指数分离性及共振数下界的存在。
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