An effective estimate for the Lebesgue measure of preimages of iterates of the Farey map

On the asymptotics of the $α$-Farey transfer operator

We study the asymptotics of iterates of the transfer operator for non-uniformly hyperbolic $α$-Farey maps. We provide a family of observables which are Riemann integrable, locally constant and of bounded variation, and for which the iterates of the transfer operator, when applied to one of these observables, is not asymptotic to a constant times the wandering rate on the first element of the partition $α$. Subsequently, sufficient conditions on observables are given under which this expected asymptotic holds. In particular, we obtain an extension theorem which establishes that, if the asymptot…

仅存在一个 Farey 映射

There is only one Farey map

本文证明在任意维度中，将标准单纯形一分为二且生成拓扑收缩投影迭代函数系统的非负矩阵对仅有三种，对应三种不同的连分数算法，其中连续情形唯一对应 Farey-Monkemeyer 映射。

Distribution of points with prescribed derivative in polynomial dynamics

In analogy to the equidistribution of preimages of a prescribed point by the iterates of a polynomial map in the complex plane towards the equilibrium measure, we show here the equidistribution of points for which the derivative of the n-th iterate of the polynomial takes a suitable prescribed value towards the equilibrium measure. We then give a similar statement in the space of degree d polynomials for the equidistribution of parameters for which the n-derivative at a given critical value has a prescribed derivative towards the activity current of the corresponding critical point.

Equilibrium States for Expanding Thurston Maps

In this paper, we use the thermodynamical formalism to show that there exists a unique equilibrium state $μ_ϕ$ for each expanding Thurston map $f: S^2\rightarrow S^2$ together with a real-valued Hölder continuous potential $ϕ$. Here the sphere $S^2$ is equipped with a natural metric induced by $f$, called a visual metric. We also prove that identical equilibrium states correspond to potentials which are co-homologous upto a constant, and that the measure-preserving transformation $f$ of the probability space $(S^2,μ_ϕ)$ is exact, and in particular, mixing and ergodic. Moreover, we establish ve…

超越函数的无限熵等分布测度

Equidistribution Measures of infinite entropy for Transcendental Functions

本文为超越整函数构造了无限熵不变测度，通过嵌入符号动力系统和传递算子两种方法，证明原像及周期点等分布，并指出测度非唯一。