Persistent Markov partitions for rational maps

Non-persistently recurrent points, qc-surgery and instability of rational maps with totally disconnected Julia sets

Let $ R $ be a rational map with totally disconnected Julia set $ J(R). $ If the postcritical set on $ J(R) $ contains a non-persistently recurrent (or conical) point, then we show that the map $ R $ can not be a structurally stable map.

Persistent Markov partitions and hyperbolic components of rational maps

Markov partitions persisting in a neighbourhood of hyperbolic components of rational maps were constructed under the condition that closures of Fatou components are disjoint in \cite{R1}. Given such a partition, we characterize all nearby hyperbolic components in terms of the symbolic dynamics. This means we can count them, and also obtain topological information. We also determine extra conditions under which all nearby type IV hyperbolic components are given by matings. These are probably the first known results of this type.

具有三个分支点的某些有理映射的可实现性

A realizability criterion for rational maps with three branch points

本文研究具有三个分支点的有理映射的Hurwitz存在性问题，通过足球分解方法建立了若干新的可实现分支数据族并识别了未记录的例外数据。

Rational points on non-linear horocycles and pigeonhole statistics for the fractional parts of $\sqrt{n}$

In this paper we investigate \textit{pigeonhole statistics} for the fractional parts of the sequence $\sqrt{n}$. Namely, we partition the unit circle $ \mathbb{T} = \mathbb{R}/\mathbb{Z}$ into $N$ intervals and show that the proportion of intervals containing exactly $j$ points of the sequence $(\sqrt{n} + \mathbb{Z})_{n=1}^N$ converges in the limit as $N \to \infty$. More generally, we investigate how the limiting distribution of the first $sN$ points of the sequence varies with the parameter $s \geq 0$. A natural way to examine this is via point processes - random measures on $[0,\infty)$ wh…

Lee-Yang-Fisher zeros for DHL and 2D rational dynamics, II. Global Pluripotential Interpretation

In a classical work of the 1950's, Lee and Yang proved that for fixed nonnegative temperature, the zeros of the partition functions of a ferromagnetic Ising model always lie on the unit circle in the complex magnetic field. Zeros of the partition function in the complex temperature were then considered by Fisher, when the magnetic field is set to zero. Limiting distributions of Lee-Yang and of Fisher zeros are physically important as they control phase transitions in the model. One can also consider the zeros of the partition function simultaneously in both complex magnetic field and complex t…