实三次映射的一个单调性猜想

关于实三次映射的熵与单调性

On Entropy and Monotonicity for Real Cubic Maps

人们已经知道拓扑熵是实二次族中参数的非减函数，这与更大的非线性导致更复杂的动力学行为的直观想法相对应。高次多项式族依赖于几个参数，因此单调性问题需要重新表述。例如，如果等熵图或具有相同拓扑熵的映射集是连通的，则可以说多参数族中的熵是单调的。本文将实三次族中等熵线的连通性问题归结为关于泛双曲性的Fatou猜想的一个弱形式。赫克曼我们还开发了一些工具，这可能被证明是有用的其他参数化的家庭，特别是一般的单调性结果发育不良的mosquitos映射：发育不良的mosquitos家庭的一个给定的形状可以被理解为一个简单的家庭，实现所有可能的组合结构，人们可以期望与一个地图的这种形状的基础上揉理论。粗略地说，我们关于实三次映射族的主要结果是，由于Heckman的结果（三次映射有两种可能的形状，取决于无穷远处的行为），实三次映射族与具有相同形状的发育三次映射族一样单调。

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.

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…

Monotonicity of entropy for unimodal real quadratic rational maps

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.

Entropy monotonicity and superstable cycles for the quadratic family revisited

The main result of this paper is a proof using real analysis of the monotonicity of the topological entropy for the family of quadratic maps, sometimes called Milnor's Monotonicity Conjecture. In contrast, the existing proofs rely in one way or another on complex analysis. Our proof is based on tools and algorithms previously developed by the authors and collaborators to compute the topological entropy of multimodal maps. Specifically, we use the number of transverse intersections of the map iterates with the so-called critical line. The approach is technically simple and geometrical. The same…

On the non-monotonicity of entropy for a class of real quadratic rational maps

We prove that the entropy function on the moduli space of real quadratic rational maps is not monotonic by exhibiting a continuum of disconnected level sets. This entropy behavior is in stark contrast with the case of polynomial maps, and establishes a conjecture on the failure of monotonicity for bimodal real quadratic rational maps of shape $(+-+)$ which was posed in arXiv:1901.03458 based on experimental evidence.