Magnus Aspenberg
In this paper we investigate the perturbation properties of rational Misiurewicz maps, when the Julia set is the whole sphere (the other case is treated in [1]). In particular, we show that if f is a Misiurewicz map and not a flexible Lattes map, then we can find a hyperbolic map arbitrarily close to f.
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We show that the set of Misiurewicz maps has Lebesgue measure zero in the parameter space of rational maps for any fixed degree greater than or equal to 2.
We show that Misiurewicz maps for which the Julia set is not the whole sphere are Lebesgue density points of hyperbolic maps.
We show that the set of Misiurewicz maps has Lebesgue measure zero in the space of rational functions for any fixed degree greater than or equal to 2 (generalising the earlier version math.DS/0701382).
We consider skew-products of quadratic maps over certain Misiurewicz-Thurston maps and study their statistical properties. We prove that, when the coupling function is a polynomial of odd degree, such a system admits two positive Lyapunov exponents almost everywhere and a unique absolutely continuous invariant probability measure.
For exponential mappings such that the orbit of the only singular value 0 is bounded, it is shown that no integrable density invariant under the dynamics exists on the complex plane.
We show that if there exists a counter example for the rational case of the Franks-Misiurewicz conjecture, then it must exhibit unbounded deviations in the complementary direction of its rotation set.
This paper continues discussions in the author's previous paper about the Misiurewicz polynomials defined for a family of degree $d \ge 2$ rational maps with an automorphism group containing the cyclic group of order $d$. In particular, we extend the sufficient conditions that the Misiurewicz polynomials are irreducible over $\mathbb{Q}$. We also prove that the Misiurewicz polynomials always have an irreducible factor of large degree.
In one-dimensional real and complex dynamics, a map whose post-singular (or post-critical) set is bounded and uniformly repelling is often called a Misiurewicz map. In results hitherto, perturbing a Misiurewicz map is likely to give a non-hyperbolic map, as per Jakobson's Theorem for unimodal interval maps. This is despite genericity of hyperbolic parameters (at least in the interval setting). We show the contrary holds in the complex exponential family $z \mapsto λ\exp(z)$: Misiurewicz maps are Lebesgue density points for hyperbolic parameters. As a by-product, we also show that Lyapunov expo…
Similarity at Misiurewicz Maps in the Cubic Parameter Curves
本研究证明了Bonifant与Milnor提出的三次参数曲线连通轨迹在Misiurewicz参数处与对应填充Julia集相似的猜想，证明思路与Kawahira推广Tan Lei的Mandelbrot集相似性证明的方法并行。
We define a hyperbolic renormalizations suitable for maps of small determinant, with uniform bounds for large periods. The techniques involve an improvement of the celebrated Palis-Takens renormalization and normal forms (fibered linearizations). These techniques are useful to study the dynamics of Hénon like maps and the geometry of their parameter space.
Misiurewicz points and subhyperbolicity in unicritical algebraic correspondences
本文首次定义了单临界代数对应关系中Misiurewicz参数，并证明在这些参数处，对应关系在Julia集邻域内均匀展开标准orbifold度量。
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.
Given a family of polynomial-like maps of large topological degree, we relate the presence of Misiurewicz parameters to a growth condition of the postcritical volume. This allows us to generalize to this setting the theory of stability and bifurcation developed by Berteloot, Dupont and the author for endomorphisms of $\mathbb{P}^k$.
We consider the family of holomorphic maps e^z+c and show that fibers of postcritically finite parameters are trivial. This can be seen as the first and simplest class of non-escaping parameters for which we can obtain results about triviality of fibers in the exponential family.
By using a suitable Banach space on which we let the transfer operator act, we make a detailed study of the ergodic theory of a unimodal map $f$ of the interval in the Misiurewicz case. We show in particular that the absolutely continuous invariant measure $ρ$ can be written as the sum of 1/square root spikes along the critical orbit, plus a continuous background. We conclude by a discussion of the sense in which the map $f\mapstoρ$ may be differentiable.
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…
The behavior under iteration of the critical points of polynomial maps plays an essential role in understanding its dynamics. We study the special case where the forward orbits of the critical points are finite. Thurston's theorem tells us that fixing a particular critical portrait and degree leads to only finitely may possible polynomials (up to equivalence) and that, in many cases, their defining equations intersect transversely. We provide explicit algebraic formulae for the parameters where the critical points of the unicritical polynomials and bicritical cubic polynomials have a specified…
The Julia set of the exponential family $E_κ:z\mapstoκe^z$, $κ>0$ was shown to be the entire complex plane when $κ>1/e$ essentially by Misiurewicz. Later, Devaney and Krych showed that for $0<κ\leq1/e$ the Julia set is an uncountable union of pairwise disjoint simple curves tending to infinity. Bergweiler generalized the result of Devaney and Krych for a three dimensional analogue of the exponential map called the Zorich map. We show that the Julia set of certain Zorich maps with symmetry is the entire $\mathbb{R}^3$ generalizing Misiurewicz's result. Moreover, we show that the periodic points…
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.
Strange attractors for the generalized Lozi-like family
本文推广 2017 年提出的类 Lozi 映射族，证明其涵盖多种 Lozi 映射及二维边界碰撞正规形，并存在作为同宿类出现的奇怪吸引子。
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.
Strange attractors for the family of orientation preserving Lozi maps
本研究将反向定向Lozi映射族存在奇异吸引子的结论推广至保定向情形，严格确定了对应参数开集，证明该区域内吸引子存在、具混沌性、为极大吸引子且在Hausdorff度量下随参数连续变化。
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。
We introduce the notion of a one-way horseshoe and show that the polynomial entropy of an interval map is given by one-way horseshoes of iterates of the map, obtaining in such a way an analogue of Misiurewicz's theorem on topological entropy and standard `two-way' horseshoes. Moreover, if the map is of Sharkovskii type 1 then its polynomial entropy can also be computed by what we call chains of essential intervals. As a consequence we get a rigidity result that if the polynomial entropy of an interval map is finite, then it is an integer. We also describe the possible values of polynomial entr…
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 …
In this article, we show that every rational map whose multipliers all lie in a given number field is a power map, a Chebyshev map or a Lattès map. This strengthens a conjecture by Milnor concerning rational maps with integer multipliers, which was recently proved by Ji and Xie.
We will prove that the zeta function for Ruelle-expanding maps is rational.
This is a preliminary investigation of the geometry and dynamics of rational maps with only two critical points. (originally titled ``On Bicritical Rational Maps'' in September 1997; revised and retitled April 1999)
We prove that the hyperbolic components of bicritical rational maps having two distinct attracting cycles each of period at least two are bounded in the moduli space of bicritical rational maps. Our arguments rely on arithmetic methods.
We give a complete description for the dynamics of quadratic rational maps with coefficients in the completion of the field of formal Puiseux series.
We study the dimension spectrum of Lyapunov exponents for rational maps on the Riemann sphere.
If $R$ is a rational map, the Main Result is a uniformization Theorem for the space of decompositions of the iterates of $R$. Secondly, we show that Fatou conjecture holds for decomposable rational maps.
Asymptotics of Transversality in Periodic Curves of Quadratic Rational Maps
本文计算了具有周期标记临界点的二次有理映射模空间的欧拉特征数，从而研究其横截性的渐近行为。
In this paper, we construct geometrically finite rational maps with buried critical points on the boundaries of some hyperbolic components by using the pinching and plumbing deformations.
Blow-up of multipliers in meromorphic families of rational maps
本文研究了黎曼球面上有理映射的单参数全纯退化族中周期循环乘子的爆破现象。
In this article, we prove that every unicritical polynomial map that has only rational multipliers is either a power map or a Chebyshev map. This provides some evidence in support of a conjecture by Milnor concerning rational maps whose multipliers are all integers.
Rational maps with constant Thurston pullback mapping
本文研究具有常值Thurston拉回映射的CTP映射，证明所有正则或混合CTP多项式满足McMullen条件，并构造了一类新的CTP映射例子。
Wandering Julia components of cubic rational maps
本文证明了三次有理映射的每个游荡 Julia 分量最终至多有两个互补分量。
Let ${\cal H}$ be a hyperbolic component of quadratic rational maps possessing two distinct attracting cycles. We show that ${\cal H}$ has compact closure in moduli space if and only if neither attractor is a fixed point.
We show that the topological entropy is monotonic for unimodal interval maps which are obtained from the restriction of quadratic rational maps with real coefficients. This is done by ruling out the existence of certain post-critical curves in the moduli space of aforementioned maps, and confirms a conjecture made in [Fil19] based on experimental evidence.
In this paper, we study hyperbolic rational maps with finitely connected Fatou sets. We construct models of post-critically finite hyperbolic tree mapping schemes for such maps, generalizing post-critically finite rational maps in the case of connected Julia set. We show they are general limits of rational maps as we quasiconformally stretch the dynamics. Conversely, we use quasiconformal surgery to show any post-critically finite hyperbolic tree mapping scheme arises as such a limit. We construct abundant examples thanks to the flexibilities of the models, and use them to construct a sequence…
In this short note we observe that within the family of slowly recurrent rational maps on the Riemann sphere, the Collet-Eckmann, second Collet-Eckmann, and topological Collet-Eckmann conditions are equivalent and also invariant under topological conjugacy.
We develop a Thurston-like theory to characterize geometrically finite rational maps, then apply it to study pinching and plumbing deformations of rational maps. We show that in certain conditions the pinching path converges uniformly and the quasiconformal conjugacy converges uniformly to a semi-conjugacy from the original map to the limit. Conversely, every geometrically finite rational map with parabolic points is the landing point of a pinching path for any prescribed plumbing combinatorics.
In this paper, we give a family of rational maps whose Julia sets are Cantor circles and show that every rational map whose Julia set is a Cantor set of circles must be topologically conjugate to one map in this family on their corresponding Julia sets. In particular, we give the specific expressions of some rational maps whose Julia sets are Cantor circles, but they are not topologically conjugate to any McMullen maps on their Julia sets. Moreover, some non-hyperbolic rational maps whose Julia sets are Cantor circles are also constructed.
A construction is given of Markov partitions for some rational maps, which persist over regions of parameter space, not confined to single hyperbolic components. The set on which the Markov partition exists, and its boundary, are analysed.
We discuss rescaling limits for sequences of complex rational maps in one variable which approach infinity in parameter space.It is shown that any given sequence of maps of degree $d \ge 2$ has at most $2d-2$ dynamically distinct rescaling limits which are not postcritically finite. For quadratic rational maps, a complete description of the possible rescaling limits is given. These results are obtained employing tools from non-Archimedean dynamics.
We present simple examples of rational maps of the complex projective plane with equal first and second dynamical degrees and no invariant foliation.
We give a complete characterization of degree two rational maps with potential good reduction over local fields. We show this happens exactly when the map corresponds to an integral point in the moduli space. We detail an algorithm by which to conjugate any degree two rational map corresponding to an integral point in the moduli space into a map with unit resultant. The local fields result is used to solve the same problem for fields over a principal ideal domain. Some additional results are given for degree 2 rational maps over the rationals.
This paper deepens the connections between critically finite rational maps and finite subdivision rules. The main theorem is that if f is a critically finite rational map with no periodic critical points, then for any sufficiently large integer n the iterate f^n is the subdivision map of a finite subdivision rule.
We study the dimension spectrum for Lyapunov exponents for rational maps acting on the Riemann sphere and characterize it by means of the Legendre-Fenchel transform of the hidden variational pressure. This pressure is defined by means of the variational principle with respect to non-atomic invariant probability measures and is associated to certain $σ$-finite conformal measures. This allows to extend previous results to exceptional rational maps.
The moduli space ${\rm M}_{d}$, of complex rational maps of degree $d \geq 2$, is a connected complex orbifold which carries a natural real structure, coming from usual complex conjugation. Its real points are the classes of rational maps admitting antiholomorphic automorphisms. The locus of the real points ${\rm M}_{d}({\mathbb R})$ decomposes as a disjoint union of the loci ${\rm M}_{d}^{\mathbb R}$, consisting of the real rational maps, and ${\mathcal P}_{d}$, consisting of the pseudo-real ones. We obtain that, both ${\rm M}_{d}^{\mathbb R}$ and ${\rm M}_{d}({\mathbb R})$, are connected and…
We discuss the dynamical, topological, and algebraic classification of rational maps $f$ of the Riemann sphere to itself each of whose critical points $c$ is also a fixed-point of $f$, i.e. $f(c)=c$.
In this paper we will give a short and elementary proof that critical relations unfold transversally in the space of rational maps.
The aim of this paper is to show $J$-stability of immediately expanding rational maps over an algebraically closed, complete, and non-Archimedean field, which is an analogue of R. Mane, P. Sad, and D. Sullivan's theorem of $J$-stability in complex dynamical systems.
A partial description of the parameter space of rational maps of degree two: Part 2
这继续研究二次有理映射族中动态变化的组合模型，重点关注其中一个临界点是周期性的那些变体。我们证明了非有理临界有限度二分支覆盖的一些一般性结果，最后确定了组合模型中有理映射的边界，从而完成了第一部分中结果的证明.
We provide a natural canonical decomposition of postcritically finite rational maps with non-empty Fatou sets based on the topological structure of their Julia sets. The building blocks of this decomposition are maps where all Fatou components are Jordan disks with disjoint closures (Sierpiński maps), as well as those where any two Fatou components can be connected through a countable chain of Fatou components with common boundary points (crochet or Newton-like maps). We provide several alternative characterizations for our decomposition, as well as an algorithm for its effective computation. …
Remarks on quadratic rational maps
这将是二次有理映射的一个简要描述。第2节到第6节是关于这类映射的几何和拓扑。第7- 10节综述了二次有理映射动力学中的一些问题。证据很少。第9节试图通过复杂的一维切片来探索和描绘模空间。第10节描述实二次有理映射的理论。为了便于说明，一些技术细节被归入附录：附录A概述了一些经典代数。附录B描述了度为\[d\]的有理映射空间的拓扑。附录C概述了二次有理映射的几种方便的标准形，并计算了各种不变量之间的关系。\break附录D描述了与曲线\[\Per_n(μ)\subset\M\]相关的一些几何形状。附录E描述了不含临界点的全不连通的朱利亚集。最后，与谭磊合作编写的附录F描述了一个连通二次朱利亚集的例子，其中补集的两个分量没有公共边界点。
In this paper, we prove that a postcritically finite rational map with non-empty Fatou set is Thurstion equivalent to an expanding Thurston map if and only if its Julia set is homeomorphic to the standard Sierpinski carpet
We analyze the infinitesimal effect of holomorphic perturbations of the dynamics of a structurally stable rational map on a neighborhood of its Julia set. This implies some restrictions on the behavior of critical points.
In this article, we show that all admissible rational maps with fixed or period two cluster cycles can be constructed by the mating of polynomials. We also investigate the polynomials which make up the matings that construct these rational maps. In the one cluster case, one of the polynomials must be an $n$-rabbit and in the two cluster case, one of the maps must be either $f$, a "double rabbit", or $g$, a secondary map which lies in the wake of the double rabbit $f$. There is also a very simple combinatorial way of classifiying the maps which must partner the aforementioned polynomials to cre…
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