arXiv math/04080112004-08math.DSmath.CV

Combinatorics of Bifurcations in Exponential Parameter Space

Lasse Rempe, Dierk Schleicher

备注：: 48 pages, 5 figures. V2: The article (particularly Section 6 and 7) was revised to improve the exposition; some figures were added. This may have changed the numbers of references to this article in other papers. In this case, please refer to the previous version of the article
发表信息：: In: Transcendental dynamics and complex analysis. In honour of Noel Baker (Rippon and Stallard, eds); London Mathematical Society Lecture Note Series 348, 317-370 (2008).

英文摘要：

We give a complete combinatorial description of the bifurcation structure in the space of exponential maps $z\mapsto\exp(z)+κ$. This combinatorial structure is the basis for a number of important results about exponential parameter space. These include the fact that every hyperbolic component has connected boundary, a classification of escaping parameters, and the fact that all dynamic and parameter rays at periodic addresses land.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Continued fractions with SL(2, Z)-branches: combinatorics and entropy

We study the dynamics of a family K_alpha of discontinuous interval maps whose (infinitely many) branches are Moebius transformations in SL(2, Z), and which arise as the critical-line case of the family of (a, b)-continued fractions. We provide an explicit construction of the bifurcation locus E_KU for this family, showing it is parametrized by Farey words and it has Hausdorff dimension zero. As a consequence, we prove that the metric entropy of K_alpha is analytic outside the bifurcation set but not differentiable at points of E_KU, and that the entropy is monotone as a function of the parame…

2013-12-24math.DS

Distribution of postcritically finite polynomials iii: Combinatorial continuity

In the first part of the present paper, we continue our study of distribution of postcritically finite parameters in the moduli space of polynomials: we show the equidistribution of Misiurewicz parameters with prescribed combinatorics toward the bifurcation measure. Our results essentially rely on a combinatorial description of the escape locus and of the bifurcation measure developped by Kiwi and Dujardin-Favre. In the second part of the paper, we construct a bifurcation measure for the connectedness locus of the quadratic anti-holomorphic family which is supported by a strict subset of the b…

2016-02-02math.CVmath.DS

Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity

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 …

2008-05-12math.CVmath.DS

Bifurcations in the Space of Exponential Maps

This article investigates the parameter space of the exponential family $z\mapsto \exp(z)+κ$. We prove that the boundary (in $\C$) of every hyperbolic component is a Jordan arc, as conjectured by Eremenko and Lyubich as well as Baker and Rippon. In fact, we prove the stronger statement that the exponential bifurcation locus is connected in $\C$, which is an analog of Douady and Hubbard's celebrated theorem that the Mandelbrot set is connected. We show furthermore that $\infty$ is not accessible through any nonhyperbolic ("queer") stable component. The main part of the argument consists of demo…

2007-10-28math.CVmath.DS

具有指数时间核的神经场动力学

Dynamics of neural fields with exponential temporal kernel

本文研究了带指数时间核的标准神经场方程，通过动态分岔分析发现其不允许静态分岔，但能产生行波等时空非恒定解，并给出了显式分岔条件。

2024-03-26math.DSnlin.PS

关于指数小分裂的几何方法：二重泛型零霍普夫分岔

A geometric approach to exponentially small splitting: The generic zero-Hopf bifurcation of co-dimension two

Technical University of Denmark（丹麦技术大学）

本文通过几何方法研究二重泛型零霍普夫分岔的展开，证明了指数小分裂的存在，并将之与广义鞍节点中心流形的非解析性联系起来，采用吹胀方法处理不同量级的动力学。

2026-04-17math.DS

更多相关论文收起更多论文

Distribution of postcritically finite polynomials

We prove that Misiurewicz parameters with prescribed combinatorics and hyperbolic parameters with (d - 1) distinct attracting cycles with given multipliers are equidistributed with respect to the bifurcation measure in the moduli space of degree d complex polynomials. Our proof relies on Yuan's equidistribution results of points of small heights, and uses in a crucial way Epstein's transversality results.

2013-02-04math.CVmath.DSmath.NT

The full renormalization horseshoe for unimodal maps of higher degree: exponential contraction along hybrid classes

We prove exponential contraction of renormalization along hybrid classes of infinitely renormalizable unimodal maps (with arbitrary combinatorics), in any even degree $d$. We then conclude that orbits of renormalization are asymptotic to the full renormalization horseshoe, which we construct. Our argument for exponential contraction is based on a precompactness property of the renormalization operator ("beau bounds"), which is leveraged in the abstract analysis of holomorphic iteration. Besides greater generality, it yields a unified approach to all combinatorics and degrees: there is no need …

2010-05-26math.DS

Distribution of postcritically finite polynomials II: Speed of convergence

In the moduli space of degree d polynomials, we prove the equidistribution of postcritically finite polynomials toward the bifurcation measure. More precisely, using complex analytic arguments and pluripotential theory, we prove the exponential speed of convergence for C 2-observables. This improves results obtained with arithmetic methods by Favre and Rivera-Letellier in the unicritical family and Favre and the first author in the space of degree d polynomials. We deduce from that the equidistribution of hyperbolic parameters with (d -- 1) distinct attracting cycles of given multipliers towar…

2016-12-02math.CVmath.DS

通过加法组合学实现指数混合

Exponential Mixing Via Additive Combinatorics

本文利用加法组合学方法构造各向异性Banach空间并证明本质谱隙，从而建立负曲率几何有限局部对称空间上测地流关于Bowen-Margulis-Sullivan测度的指数混合性，同时给出相关函数Laplace变换的亚纯延拓结果。

2024-12-28math.CAmath.COmath.DS

Capturing the cascade: a transseries approach to delayed bifurcations

Transseries expansions build upon ordinary power series methods by including additional basis elements such as exponentials and logarithms. Alternative summation methods can then be used to "resum" series to obtain more efficient approximations, and have been successfully widely applied in the study of continuous linear and nonlinear, single and multidimensional problems. In particular, a method known as transasymptotic resummation can be used to describe continuous behaviour occurring on multiple scales without the need for asymptotic matching. Here we apply transasymptotic resummation to dis…

2020-12-17math-phmath.DSmath.MP

A Combinatorial Classification of Postsingularly Finite Complex Exponential Maps

We give a combinatorial classification of postsingularly finite exponential maps in terms of external addresses starting with the entry 0. This is an extension of the classification results for critically preperiodic polynomials \cite{BFH} to exponential maps. Our proof relies on the topological characterization of postsingularly finite exponential maps given recently in \cite{HSS}. Our results illustrate once again the fruitful interplay between combinatorics, topology and complex structure which has often been successful in complex dynamics.

2007-05-27math.DS

Delayed Hopf bifurcation and space-time buffer curves in the Complex Ginzburg-Landau equation

In this article, the phenomenon of delayed Hopf bifurcations (DHB) in reaction-diffusion PDEs is analyzed in the cubic Complex Ginzburg-Landau equation with a slowly-varying parameter. We use the classical asymptotic methods of stationary phase and steepest descents to show that solutions which approach the attracting quasi-steady state (QSS) before the Hopf bifurcation remain near that state for long times after the Hopf bifurcation and the QSS has become repelling. In the complex time plane, the phase function of the linear PDE has a saddle point, and the Stokes and anti-Stokes lines are cen…

2020-12-18math.DS

椭圆方程在多维领域上的指数分裂

Exponential Dichotomies for Elliptic Equations on Multidimensional Domains

Boston University（波士顿大学）

本文研究了多维领域上椭圆方程的指数分裂存在性，通过空间动力学方法扩展至一般多维空间域，为相干结构的未来分析提供工具。

2026-08-03math.APmath.DS

A meeting point of entropy and bifurcations in cross-diffusion herding

A cross-diffusion system modeling the information herding of individuals is analyzed in a bounded domain with no-flux boundary conditions. The variables are the species' density and an influence function which modifies the information state of the individuals. The cross-diffusion term may stabilize or destabilize the system. Furthermore, it allows for a formal gradient-flow or entropy structure. Exploiting this structure, the global-in-time existence of weak solutions and the exponential decay to the constant steady state is proved in certain parameter regimes. This approach does not extend to…

2016-03-01math.APmath.DSnlin.PS

Dynamics of exponential linear map in functional space

We consider the question of existence of a unique invariant probability distribution which satisfies some evolutionary property. The problem arises from the random graph theory but to answer it we treat it as a dynamical system in the functional space, where we look for a global attractor. We consider the following bifurcation problem: Given a probability measure $μ$, which corresponds to the weight distribution of a link of a random graph we form a positive linear operator $Φ$ (convolution) on distribution functions and then we analyze a family of its exponents with a parameter $λ$ which corr…

2004-02-21math.DSmath.PR