Rational Misiurewicz maps are rare II

Misiurewicz polynomials for rational maps with nontrivial automorphisms

In this paper, we consider a one-parameter family of degree $d\ge 2$ rational maps with an automorphism group containing the cyclic group of order $d$. We construct a polynomial whose roots correspond to parameter values for which the corresponding map is post-critically finite with a certain dynamical portrait. Then we prove that the polynomial is irreducible in certain cases.

Misiurewicz点处Multibrot集与对应关系的Julia集之间的相似性

Similarity between the Multibrot set and the Julia set of correspondences at Misiurewicz points

本研究证明在Misiurewicz点处，代数对应关系的Julia集渐近自相似，且Multibrot集与Julia集渐近相似，并给出横截性条件的代数证明及实验支持。

Inducing techniques for quantitative recurrence and applications to Misiurewicz maps and doubly intermittent maps

We prove an abstract result establishing that one can obtain the convergence of Rare Events Point Processes counting the number of orbital visits to a sequence of shrinking target sets from the convergence of corresponding point processes for some induced system and matching shadowing shrinking sets inside the base of the inducing scheme. We apply this result to prove a dichotomy for two classes of non-uniformly hyperbolic interval maps: Misiurewicz quadratic maps and doubly intermittent maps. The dichotomy holds in the sense that the shrinking target sets may accumulate in any individual poin…

Rational Parameter Rays of The Multibrot Sets

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…

Laplacians on Julia sets of rational maps

The study of Julia sets gives a new and natural way to look at fractals. When mathematicians investigated the special class of Misiurewicz's rational maps, they found out that there is a Julia set which is homeomorphic to a well known fractal, the Sierpinski gasket. In this paper, we apply the method of Kigami to give rise to a new construction of Laplacians on the Sierpinski gasket like Julia sets with a dynamically invariant property.

Wild recurrent critical points

It is conjectured that a rational map whose coefficients are algebraic over $\Q_p$ has no wandering components of the Fatou set. R. Benedetto has shown that any counter example to this conjecture must have a wild recurrent critical point. We provide here the first examples of rational maps whose coefficients are algebraic over $\Q_p$ and that have a (wild) recurrent critical point. In fact, we show that there is such a rational map in every one parameter family of rational maps that is defined over a finite extension of $\Q_p$ and that has a Misiurewicz bifurcation.

Distribution of rational maps with a preperiodic critical point

Let {f_t} be any algebraic family of rational maps of a fixed degree, with a marked critical point c(t). We first prove that the hypersurfaces of parameters for which c(t) is periodic converge as a sequence of positive closed (1,1) currents to the bifurcation current attached to c and defined by DeMarco. We then turn our attention to the parameter space of polynomials of a fixed degree d. By intersecting the d-1 currents attached to each critical point of a polynomial, Bassaneli and Berteloot obtained a positive measure of finite mass which is supported on the connectedness locus. They showed …

On annular maps of the torus and sublinear diffusion

There is a classification by Misiurewicz and Ziemian of elements in Homeo$_0(\mathbf{T}^2)$ by their rotation set $ρ$, according to wether $ρ$ is a point, a segment or a set with nonempty interior. A recent classification of nonwandering elements in Homeo$_0(\mathbf{T}^2)$ by Koropecki and Tal has been given, according to the itrinsic underlying ambient where the dynamics takes place: planar, annular and strictly toral maps. We study the link between these two classifications, showing that, even abroad the nonwandering setting, annular maps are characterized by rotation sets which are \textit{…

手术序列与曼德博集合的自相似性

Surgery sequences and self-similarity of the Mandelbrot set

本文在有理映射中引入双曲德恩手术的类比，定义手术序列，并利用其给出谭蕾关于朱利亚集与曼德博集合在米修雷维奇点处渐近自相似性定理的新初等证明。

Semi-hyperbolic maps are rare

We prove in this paper that the set of semi-hyperbolic rational maps has Lebesgue measure zero in the space of rational maps of the Riemann sphere for a fixed degree d at least 2. It generalises an earlier result by J. Graczyk and the author who proved the same thing in degree 2.

具有周期临界轨道的三次多项式映射，第三部分：镶嵌与轨道肖像

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。

扩张Thurston映射的基态与周期轨道

Ground states and periodic orbits for expanding Thurston maps

本文研究扩张Thurston映射的最大化测度与基态，证明典型周期优化猜想成立，并推广到Misiurewicz–Thurston有理映射。