Core entropy and biaccessibility of quadratic polynomials

Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set

Let c be a real parameter in the Mandelbrot set, and f_c(z):= z^2 + c. We prove a formula relating the topological entropy of f_c to the Hausdorff dimension of the set of rays landing on the real Julia set, and to the Hausdorff dimension of the set of rays landing on the real section of the Mandelbrot set, to the right of the given parameter c. We then generalize the result by looking at the entropy of Hubbard trees: namely, we relate the Hausdorff dimension of the set of external angles which land on a certain slice of a principal vein in the Mandelbrot set to the topological entropy of the q…

Monotonicity of entropy for real quadratic rational maps

The monotonicity of entropy is investigated for real quadratic rational maps on the real circle $\mathbb{R}\cup\{\infty\}$ based on the natural partition of the corresponding moduli space $\mathcal{M}_2(\mathbb{R})$ into its monotonic, covering, unimodal and bimodal regions. Utilizing the theory of polynomial-like mappings, we prove that the level sets of the real entropy function $h_\mathbb{R}$ are connected in the $(-+-)$-bimodal region and a portion of the unimodal region in $\mathcal{M}_2(\mathbb{R})$. Based on the numerical evidence, we conjecture that the monotonicity holds throughout th…

Diabolical entropy

Milnor and Thurston's famous paper proved monotonicity of the topological entropy for the real quadratic family. Guckenheimer showed that it is Hölder continuous. We obtain a precise formula for the Hölder exponent at almost every quadratic parameter. Furthermore, the entropy of most parameters is proven to be in a set of Hausdorff dimension smaller than one, while most values of the entropy arise from a set of parameters of dimension smaller than one.

二次数域中的无幂点：稳定子群、动力学与熵

Power-free points in quadratic number fields: Stabiliser, dynamics and entropy

该研究聚焦一般二次数域中的k无幂整数集，考察其扩展对称性对诱导拓扑动力系统的影响，建立数论与动力学量的对应，并用对称性和熵区分不同系统。

The core entropy for polynomials of higher degree

As defined by W. Thurston, the core entropy of a polynomial is the entropy of the restriction to its Hubbard tree. For each d >= 2, we study the core entropy as a function on the parameter space of polynomials of degree d, and prove it varies continuously both as a function of the combinatorial data and of the coefficients of the polynomials. This generalizes a conjecture of Thurston for quadratic polynomials.

Monotonicity of entropy for real multimodal maps

In \cite{Mil}, Milnor posed the {\em Monotonicity Conjecture} that the set of parameters within a family of real multimodal polynomial interval maps, for which the topological entropy is constant, is connected. This conjecture was proved for quadratic by Milnor & Thurston \cite{MT} and for cubic maps by Milnor & Tresser, see \cite{MTr} and also \cite{DGMT}. In this paper we will prove the general case.

On the Lebesgue measure of the Julia set of a quadratic polynomial

The goal of this note is to prove the following theorem: Let $p_a(z) = z^2+a$ be a quadratic polynomial which has no irrational indifferent periodic points, and is not infinitely renormalizable. Then the Lebesgue measure of the Julia set $J(p_a)$ is equal to zero. As part of the proof we discuss a property of the critical point to be {\it persistently recurrent}, and relate our results to corresponding ones for real one dimensional maps. In particular, we show that in the persistently recurrent case the restriction $p_a|ω(0)$ is topologically minimal and has zero topological entropy. The Douad…

幂律单峰映射的拓扑熵

Topological Entropy for Power-Law Unimodal Maps

本文证明了幂律单峰映射中 kneading 序列与拓扑熵的单调性，通过引入 Thurston 类型算子和确定性恒等式，解决了非整数临界性下的非多项式动态问题，展示了熵单调性在非多项式动态中的鲁棒性。

二元二次型由四元二次型表示

Representations of binary forms by quaternary quadratic forms

本文通过测度刚性结果和行列方法，证明了二元二次型由四元二次型表示的局部-全局原理。

广义抛物动力学的熵

Entropy for Generalized Parabolic Dynamics

本文将广义熵扩展至一致空间，引入广义抛物动力学类，证明其重要子类具有线性熵，并构造了紧曲面同胚实例，使其广义熵实现预设的多项式增长且非游荡集简化为单点。

Quadratic Irrationals, Closed Geodesics on the Modular Surface and Dynamical Zeta Functions

We show that generating functions associated to the sequence of convergents of a quadratic irrational are related in a natural way to the dynam- ical zeta function of a hyperbolic automorphism of the 2-torus. As a corollary, this shows that the Lévy constant of a quadratic irrational appears naturally as the topological entropy of such maps.