Topological entropy of m-fold maps on trees

Density of periodic sources in the boundary of a basin of attraction for iteration of holomorphic maps, geometric coding trees technique

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.

临界后无限的单临界多项式与指数映射的组合 Hubbard 树

Combinatorial Hubbard trees for postcritically infinite unicritical polynomials and exponential maps

本文从外部角或揉序列出发，为临界值不逃逸的所有单临界多项式与指数映射构造正向不变的组合 Hubbard 树，将经典构造推广到无限临界轨道、Julia 集非道路连通及非复容许揉序列情形。

Approximations for Gibbs states of arbitrary Holder potentials on hyperbolic folded sets

In the case of smooth non-invertible maps which are hyperbolic on folded basic sets $Λ$, we give approximations for the Gibbs states (equilibrium measures) of arbitrary Hölder potentials, with the help of weighted sums of atomic measures on preimage sets of high order. Our endomorphism may have also stable directions on $Λ$, thus it is non-expanding in general. Folding of the phase space means that we do not have a foliation structure for the local unstable manifolds (they depend on the whole past and may intersect each other both inside and outside $Λ$), and moreover the number of preimages r…

On the Class of Similar Square {-1,0,1}-Matrices Arising from Vertex maps on Trees

Let $n \ge 2$ be an integer. In this note, we show that the {\it oriented} transition matrices over the field $\mathcal R$ of all real numbers (over the finite field $\mathcal Z_2$ of two elements respectively) of all continuous {\it vertex maps} on {\it all} oriented trees with $n+1$ vertices are similar to one another over $\mathcal R$ (over $\mathcal Z_2$ respectively) and have characteristic polynomial $\sum_{k=0}^n x^k$. Consequently, the {\it unoriented} transition matrices over the field $Z_2$ of all continuous {\it vertex maps} on {\it all} oriented trees with $n+1$ vertices are simila…

Periods implying almost all periods, trees with snowflakes, and zero entropy maps

Let $X$ be a compact tree, $f$ be a continuous map from $X$ to itself, $End(X)$ be the number of endpoints and $Edg(X)$ be the number of edges of $X$. We show that if $n>1$ has no prime divisors less than $End(X)+1$ and $f$ has a cycle of period $n$, then $f$ has cycles of all periods greater than $2End(X)(n-1)$ and topological entropy $h(f)>0$; so if $p$ is the least prime number greater than $End(X)$ and $f$ has cycles of all periods from 1 to $2End(X)(p-1)$, then $f$ has cycles of all periods (this verifies a conjecture of Misiurewicz for tree maps). Together with the spectral decomposition…

Spectral gap of sparse bistochastic matrices with exchangeable rows with application to shuffle-and-fold maps

We consider a random bistochastic matrix of size $n$ of the form $M Q$ where $M$ is a uniformly distributed permutation matrix and $Q$ is a given bistochastic matrix. Under mild sparsity and regularity assumptions on $Q$, we prove that the second largest eigenvalue of $MQ$ is essentially bounded by the normalized Hilbert-Schmidt norm of $Q$ when $n$ grows large. We apply this result to random walks on random regular digraphs and to shuffle-and-fold maps of the unit interval popularized in fluid mixing protocols.

树上零熵循环：从拓扑到组合学及星图的应用

Zero entropy cycles on trees: from Topology to Combinatorics and an application to star maps

本文通过组合方法描述树上零熵周期模式，提供一种不依赖拓扑实现的快速算法。研究星图上的零熵映射，并描述其周期模式。

$C^{1+α}$映射的折叠型熵公式

Entropy formula of folding type for $C^{1+α}$ maps

本文针对一般$C^{1+α}$映射（含零Jacobian退化情形）建立了折叠型熵公式，证明测度为逆SRB测度当且仅当该公式成立且Jacobian级数可积，为此发展了相应的Pesin理论。

Approximability of dynamical systems between trees of spheres

We study sequences of analytic conjugacy classes of rational maps which diverge in moduli space. In particular, we are interested in the notion of rescaling limits introduced by Jan Kiwi. In the continuity of [A1] we recall the notion of dynamical covers between trees of spheres for which a periodic sphere corresponds to a rescaling limit. We study necessary conditions for such a dynamical cover to be the limit of dynamically marked rational maps. With these conditions we classify them for the case of bicritical maps and we recover the second main result of Jan Kiwi regarding rescaling limits.

Trees, length spectra for rational maps via barycentric extensions and Berkovich spaces

In this paper, we study the dynamics of degenerating sequences of rational maps on Riemann sphere $\hat{\mathbb{C}}$ using $\mathbb{R}$-trees. Given a sequence of degenerating rational maps, we give two constructions for limiting dynamics on $\mathbb{R}$-trees: one geometric and one algebraic. The geometric construction uses the ultralimit of rescalings of barycentric extensions of rational maps, while the algebraic construction uses the Berkovich space of complexified Robinson's field. We show the two approaches are equivalent. The limiting dynamics on the $\mathbb{R}$-tree are analogues to i…

Topological entropy and secondary folding

A convenient measure of a map or flow's chaotic action is the topological entropy. In many cases, the entropy has a homological origin: it is forced by the topology of the space. For example, in simple toral maps, the topological entropy is exactly equal to the growth induced by the map on the fundamental group of the torus. However, in many situations the numerically-computed topological entropy is greater than the bound implied by this action. We associate this gap between the bound and the true entropy with 'secondary folding': material lines undergo folding which is not homologically force…

广义伪阿诺索夫映射与哈伯德树

Generalized Pseudo-Anosov Maps and Hubbard Trees

本研究建立二次多项式与曲面同胚动力学的联系，定义无交叉型哈伯德树，并证明此类树恰为可构造广义伪阿诺索夫映射的充要条件。

Limits of rational maps, R-trees and barycentric extension

In this paper, we show that one can naturally associate a limiting dynamical system $F: T\longrightarrow T$ on an $\R$-tree to any degenerating sequence of rational maps $f_n: \hat\C \longrightarrow \hat\C$ of fixed degree. The construction of $F$ is in $2$ steps: first we use barycentric extension to get $\E f_n : \Hyp^3 \longrightarrow \Hyp^3$; second, we take appropriate limit on rescalings of hyperbolic space. An important ingredient we prove here is that the Lipschitz constant depends only on the degree of the rational map. We show that the dynamics of $F$ records the limiting length spec…