In this paper we introduce the notion of parabolic-like mapping, which is an object similar to a polynomial-like mapping, but with a parabolic external class, i.e. an external map with a parabolic fixed point. We prove a straightening theorem for parabolic-like maps, which states that any parabolic-like map of degree 2 is hybrid conjugate to a member of the family Per_1(1), and this member is unique (up to holomorphic conjugacy) if the filled Julia set of the parabolic-like map is connected.
We show that the definition of parabolic-like map can be slightly modified, by asking $\partial Δ$ to be a quasiarc out of the parabolic fixed point, instead of the dividing arcs to be $C^1$ on $[-1,0]$ and $[0,1]$.
Dynamics on parabolic immediate basins for rational Newton maps of entire functions have been studied. It is proved that every parabolic immediate basin contains invariant accesses to the parabolic fixed point at infinity. Moreover, among these accesses there exists a unique dynamically defined access where dynamics are attracted towards the parabolic fixed point, whereas for other accesses, if there is any, the dynamics are repelled.
We prove that any $C^{1+BV}$ degree $d \geq 2$ circle covering $h$ having all periodic orbits weakly expanding, is conjugate in the same smoothness class to a metrically expanding map. We use this to connect the space of parabolic external maps (coming from the theory of parabolic-like maps) to metrically expanding circle coverings.
Inou and Shishikura provided a class of maps that is invariant by near-parabolic renormalization, and that has proved extremely useful in the study of the dynamics of quadratic polynomials. We provide here another construction, using more general arguments. This will allow to extend the range of applications to unicritical polynomials of all degrees.
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.
The paper deals with Newton maps of complex exponential functions and a surgery tool developed by P. Haïssinsky. The concept of "Postcritically minimal" Newton maps of complex exponential functions are introduced, analogous to postcritically finite Newton maps of polynomials. The dynamics preserving mapping is constructed between the space of postcritically finite Newton maps of polynomials and the space of postcritically minimal Newton maps of complex exponential functions.
We prove that for a dense set of irrational numbers $α$, the analytic centraliser of the map $e^{2πi α} z+ z^2$ near $0$ is trivial. We also prove that some analytic circle diffeomorphisms in the Arnold family, with irrational rotation numbers, have trivial centralisers. These provide the first examples of such maps with trivial centralisers.
We consider parabolic flows on 3-dimensional manifolds which are renormalized by circle extensions of Anosov diffeormorphisms. This class of flows includes nilflows on the Heisenberg nilmanifold which are renormalized by partially hyperbolic automorphisms. The transfer operators associated to the renormalization maps, acting on anisotropic Sobolev spaces, are known to have good spectral properties (this relies on ideas which have some resemblance to representation theory but also apply to non-algebraic systems). The spectral information is used to describe the deviation of ergodic averages and…
In this paper we give a combinatorial description of the renormlization limits of infinitely renormalizable unimodal maps with {\it essentially bounded} combinatorics admitting quadratic-like complex extensions. As an application we construct a natural analogue of the period-doubling fixed point. Dynamical hairiness is also proven for maps in this class. These results are proven by analyzing {\it parabolic towers}: sequences of maps related either by renormalization or by {\it parabolic renormalization}.
We introduce the study of the local dynamics around a parabolic indifferent invariant curve for fibred holomorphic maps. As in the classical non-fibred case, we show that petals are the main ingredient. Nevertheless, one expects the properties of the base rotation number should play an important role in the arrangement of the petals. We exhibit examples where the existence and the number of petals depend not just on the complex coordinate of the map, but on the base rotation number. Furthermore, under additional hypothesis on the arithmetics and smoothness of the map, we present a theorem that…
In this paper we consider families of holomorphic maps defined on subsets of the complex plane, and show that the technique developed in \cite{LSvS1} to treat unfolding of critical relations can also be used to deal with cases where the critical orbit converges to a hyperbolic attracting or a parabolic periodic orbit. As before this result applies to rather general families of maps, such as polynomial-like mappings, provided some lifting property holds. Our Main Theorem states that either the multiplier of a hyperbolic attracting periodic orbit depends univalently on the parameter and bifurcat…
For a class of polynomial maps of one variable with a parabolic fixed points and degrees bigger than $21$, the parabolic renormalization is introduced based on Fatou coordinates and horn maps, and a type of maps which are invariant under the parabolic renormalization is also given. For the small perturbation of these kinds of maps, the near-parabolic renormalization is also introduced based on the first return maps defined on the fundamental regions. As an application, we show the existence of non-renormalizable polynomial maps with degrees bigger than $21$ such that the Julia sets have positi…
We provide an abstract framework for the study of certain spectral properties of parabolic systems; specifically, we determine under which general conditions to expect the presence of absolutely continuous spectral measures. We use these general conditions to derive results for spectral properties of time-changes of unipotent flows on homogeneous spaces of semisimple groups regarding absolutely continuous spectrum as well as maximal spectral type; the time-changes of the horocycle flow are special cases of this general category of flows. In addition we use the general conditions to derive spec…
Systems of non-autonomous parabolic partial differential equations over a bounded domain with nonlinear term of Carathéodory type are considered. Appropriate topologies on sets of Lipschitz Carathéodory maps are defined in order to have a continuous dependence of the mild solutions with respect to the variation of both the nonlinear term and the initial conditions, under different assumptions on the bound-maps of the nonlinearities.
We show that the tangential speed of a parabolic semigroup of holomorphic self-maps in the unit disc is asymptotically bounded from above by (1/2)logt, proving a conjecture by Bracci. In order to show the proof we need a result of "asymptotical monotonicity" of the tangential speed for proper pairs of parabolic semigroups with positive hyperbolic step.
We prove that if A is the basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, then periodic points in the boundary of A are dense in this boundary. To prove this in the non simply- connected or parabolic situations we prove a more abstract, geometric coding trees version.
Consider the parameter space $\mathcal{P}_λ\subset \mathbb{C}^{2}$ of complex Hénon maps $$ H_{c,a}(x,y)=(x^{2}+c+ay,ax),\ \ a\neq 0 $$ which have a semi-parabolic fixed point with one eigenvalue $λ=e^{2πi p/q}$. We give a characterization of those Hénon maps from the curve $\mathcal{P}_λ$ that are small perturbations of a quadratic polynomial $p$ with a parabolic fixed point of multiplier $λ$. We prove that there is an open disk of parameters in $\mathcal{P}_λ$ for which the semi-parabolic Hénon map has connected Julia set $J$ and is structurally stable on $J$ and $J^{+}$. The Julia set $J^{+…
We consider the structure of substantially dissipative complex Hénon maps admitting a dominated splitting on the Julia set. The dominated splitting assumption corresponds to the one-dimensional assumption that there are no critical points on the Julia set. Indeed, we prove the corresponding description of the Fatou set, namely that it consists of only finitely many components, each either attracting or parabolic periodic. In particular there are no rotation domains, and no wandering components. Moreover, we show that $J = J^\star$ and the dynamics on $J$ is hyperbolic away from parabolic cycle…
Institut Denis Poisson, Université d’Orléans（奥尔良大学）；Departamento de Física y Matemáticas, Universidad de Alcalá（阿尔卡拉大学）；Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251（波尔多大学）；Institut Universitaire de France (IUF)（法国高等研究院）
We study the cohomological equation for discrete horocycle maps on $SL(2, \mathbb{R})$ and $SL(2,\mathbb R)\times SL(2, \mathbb{R})$ via representation theory. Specifically, we prove Hilbert Sobolev non-tame estimates for solutions of the cohomological equation of horocycle maps in representations of $SL(2,\mathbb R)$. Our estimates improve on previous results and are sharp up to a fixed, finite loss of regularity. Moreover, they are tame on a co-dimension one subspace of $sl(2, \mathbb R)$, and we prove tame cocycle rigidity for some two-parameter discrete actions, improving on a previous res…
We prove some new continuity results for the Julia sets $J$ and $J^{+}$ of the complex Hénon map $H_{c,a}(x,y)=(x^{2}+c+ay, ax)$, where $a$ and $c$ are complex parameters. We look at the parameter space of dissipative Hénon maps which have a fixed point with one eigenvalue $(1+t)λ$, where $λ$ is a root of unity and $t$ is real and small in absolute value. These maps have a semi-parabolic fixed point when $t$ is $0$, and we use the techniques that we have developed in [RT] for the semi-parabolic case to describe nearby perturbations. We show that for small nonzero $|t|$, the Hénon map is hyperb…
Queen Mary University of London（伦敦玛丽女王大学）；Instituto de Matemática Pura e Aplicada（巴西纯粹与应用数学研究所）；Institute for Mathematical Sciences, Stony Brook University（石溪大学数学科学研究所）；Tata Institute of Fundamental Research（塔塔基础研究所）
In this paper we study analytic families of degree 2 parabolic-like mappings (as we defined in arXiv:1111.7150). We prove that the corresponding family of hybrid conjugacies induces a continuous map, which associates to each parameter the multiplier of the fixed point of the hybrid equivalent rational map P_A. We prove that, under suitable conditions, this map restricts to a ramified covering from the connectedness locus of the family of parabolic-like maps to the connectedness locus M_1 \ {1}.
We consider holomorphic maps $f: U \to U$ for a hyperbolic domain $U$ in the complex plane, such that the iterates of $f$ converge to a boundary point $ζ$ of $U$. By a previous result of the authors, for such maps there exist nice absorbing domains $W \subset U$. In this paper we show that $W$ can be chosen to be simply connected, if $f$ has parabolic I type in the sense of the Baker--Pommerenke--Cowen classification of its lift by a universal covering (and $ζ$ is not an isolated boundary point of $U$). Moreover, we provide counterexamples for other types of the map $f$ and give an exact chara…
The local dynamics around a fixed point has been extensively studied for germs of one and several complex variables. In one dimension, there exist a complete picture of the trajectory of the orbits on a whole neighborhood of the fixed point. In dimensions larger or equal than two some partial results are known. In this article we analyze a case that lies in the boundary between one and several complex variables. We consider skew product maps of the form F (z, w) = (λ(z), f (z, w)). We deal with the case of parabolic skew product maps, that is when DF(0,0) = Id. Our goal is to describe the beha…
A {\sl parabolic cylinder} is an invariant, non-recurrent Fatou component $Ω$ of an automorphism $F$ of $\mathbb C^2$ satisfying: (1) The closure of the $ω$-limit set of $F$ on $Ω$ contains an isolated fixed point, (2) there exists a univalent map $Φ$ from $Ω$ into $\mathbb C^2$ conjugating $F$ to the translation $(z,w) \mapsto (z+1, w)$, and (3) every limit map of $\{F^{\circ n}\}$ on $Ω$ has one-dimensional image. In this paper we prove the existence of parabolic cylinders for an explicit class of maps, and show that examples in this class can be constructed as compositions of shears and ove…
According to an analogy to quasi-Fuchsian groups, we investigate topological and combinatorial structures of Lyubich and Minsky's affine and hyperbolic 3-laminations associated with the hyperbolic and parabolic quadratic maps. We begin by showing that hyperbolic rational maps in the same hyperbolic component have quasi-isometrically the same 3-laminations. This gives a good reason to regard the main cardioid of the Mandelbrot set as an analogue of the Bers slices in the quasi-Fuchsian space. Then we describe the topological and combinatorial changes of laminations associated with hyperbolic-to…
We classify those rational maps $f: \mathbb{P}^1 \to \mathbb{P}^1$ for which there exists a contravariant tensor $q$ which is parallel, i.e. such that $f^*q // q$, by proving that such maps preserve a parabolic orbifold.
The planar $(n+1)$-body problem models the motion of $n+1$ bodies in the plane under their mutual Newtonian gravitational attraction forces. When $n\ge 3$, the question about final motions, that is, what are the possible limit motions in the planar $(n+1)$-body problem when $t\to \infty$, ceases to be completely meaningful due to the existence of non-collision singularities. In this paper we prove the existence of solutions of the planar $(n+1)$-body problem which are defined for all forward time and tend to a parabolic motion, that is, that one of the bodies reaches infinity with zero velocit…
The restricted planar elliptic three body problem (RPETBP) describes the motion of a massless particle (a comet) under the gravitational field of two massive bodies (the primaries, say the Sun and Jupiter) revolving around their center of mass on elliptic orbits with some positive eccentricity. The aim of this paper is to show the existence of orbits whose angular momentum performs arbitrary excursions in a large region. In particular, there exist diffusive orbits, that is, with a large variation of angular momentum. The leading idea of the proof consists in analysing parabolic motions of the …
We give an estimate of the discontinuity of the large Julia set for a perturbation of a class of maps tangent to the identity, by means of a two-dimensional Lavaurs Theorem. We adapt to our situation a strategy due to Bedford, Smillie and Ueda in the semiattracting setting. We also prove the discontinuity of the filled Julia set for such perturbations of regular polynomials.
We develop a variational principle that extends the notion of a shearless transport barrier from steady to general unsteady two-dimensional flows and maps defined over a finite time interval. This principle reveals that hyperbolic Lagrangian Coherent Structures (LCSs) and parabolic LCSs (or jet cores) are the two main types of shearless barriers in unsteady flows. Based on the boundary conditions they satisfy, parabolic barriers are found to be more observable and robust than hyperbolic barriers, confirming widespread numerical observations. Both types of barriers are special null-geodesics of…
It is well-known that the Manneville-Pomeau map with a parabolic fixed point of the form $x\mapsto x+x^{1+α} \mod 1$ is stochastically stable for $α\ge 1$ and the limiting measure is the Dirac measure at the fixed point. In this paper we show that if $α\in (0,1)$ then it is also stochastically stable. Indeed, the stationary measure of the random map converges strongly to the absolutely continuous invariant measure for the deterministic system as the noise tends to zero.
We introduce a notion of hyperbolicity and parabolicity for a holomorphic self-map $f: Δ^N \to Δ^N$ of the polydisc which does not admit fixed points in $Δ^N$. We generalize to the polydisc two classical one-variable results: we solve the Valiron equation for a hyperbolic $f$ and the Abel equation for a parabolic nonzero-step $f$. This is done by studying the canonical Kobayashi hyperbolic semi-model of $f$ and by obtaining a normal form for the automorphisms of the polydisc. In the case of the Valiron equation we also describe the space of all solutions.
By Dulac maps we mean first return maps of hyperbolic polycycles of analytic planar vector fields. We study the fractal properties of the orbits of a parabolic Dulac map. To this end, we prove that it admits a Fatou coordinate with an asympotic expansion in terms of power-iterated logarithm transseries. This allows to introduce a new notion, the \emph{continuous time length of $\varepsilon$-neighborhoods of orbits}, and to prove that this function of $\varepsilon$ admits an asymptotic expansion in the same scale. We show that, under some hypotheses, this expansion determines the class of forma…
In this paper we introduce and study a certain intricate Cantor-like set $C$ contained in unit interval. Our main result is to show that the set $C$ itself, as well as the set of dissipative points within $C$, both have Hausdorff dimension equal to 1. The proof uses the transience of a certain non-symmetric Cauchy-type random walk.
Let $f(z) = z+z^2+O(z^3)$ and $f_ε(z) = f(z) + ε^2$. A classical result in parabolic bifurcation in one complex variable is the following: if $N-\fracπε\to 0$ we obtain $(f_ε)^{N} \to \mathcal{L}_f$, where $\mathcal{L}_f$ is the Lavaurs map of $f$. In this paper we study a \textit{non-autonomous} parabolic bifurcation. We focus on the case of $f_0(z)=\frac{z}{1-z}$. Given a sequence $\{ε_i\}_{1\leq i\leq N}$, we denote $f_n(z) = f_0(z) + ε_n^2$. We give sufficient and necessary conditions on the sequence $\{ε_i\}$ that imply that $f_{N}\circ\ldots f_{1} \to \textrm{Id}$ (the Lavaurs map of $f_…
In this paper, we continue exploration of the dynamical and parameter planes of one-parameter families of Schwarz reflections that was initiated in \cite{LLMM1,LLMM2}. Namely, we consider a family of quadrature domains obtained by restricting the Chebyshev cubic polynomial to various univalent discs. Then we perform a quasiconformal surgery that turns these reflections to parabolic rational maps (which is the crucial technical ingredient of our theory). It induces a straightening map between the parameter plane of Schwarz reflections and the parabolic Tricorn. We describe various properties of…
