Variation of argument and Bernstein index for holomorphic functions on Riemann surfaces

关于紧致 Riemann 曲面的全纯管状邻域

On holomorphic tubular neighborhoods of compact Riemann surfaces

本文给出了全纯嵌入于非奇异复曲面中的紧致 Riemann 曲面存在全纯管状邻域的充分条件，该条件由酉平坦线丛的算术条件描述，是单复变动力系统中无理数 Brjuno 条件的类比。

The Riemann hypothesis and holomorphic index in complex dynamics

We give an interpretation of the Riemann hypothesis in terms of complex and topological dynamics. For example, the Riemann hypothesis is affirmative and all zeros of the Riemann zeta function are simple if and only if a certain meromorphic function has no attracting fixed point. To obtain this, we use holomorphic index (residue fixed point index), which characterizes local properties of fixed points in complex dynamics.

Dynamics of holomorphic correspondences on Riemann Surfaces

We study the dynamics of holomorphic correspondences $f$ on a compact Riemann surface $X$ in the case, so far not well understood, where $f$ and $f^{-1}$ have the same topological degree. Under a mild and necessary condition that we call non weak modularity, $f$ admits two canonical probability measures $μ^+$ and $μ^-$ which are invariant by $f^*$ and $f_*$ respectively. If the critical values of $f$ (resp. $f^{-1}$) are not periodic, the backward (resp. forward) orbit of any point $a \in X$ equidistributes towards $μ^+$ (resp. $μ^-$), uniformly in $a$ and exponentially fast.

Partial holomorphic semiconjugacies between rational functions

We establish a general result on the existence of partially defined semiconjugacies between rational functions acting on the Riemann sphere. The semiconjugacies are defined on the complements to at most one-dimensional sets. They are holomorphic in a certain sense.

Holomorphic Hamiltonian $ξ$-Flow and Riemann Zeros

With a view on the formal analogy between Riemann-von-Mangoldts explicit formula and semiclassical quantum mechanics in terms of the Gutzwiller trace formula we construct a complex-valued Hamiltonian $H(q,p)=ξ(q)p$ from the holomorphic flow $\dot{q}=ξ(q)$ and its variational differential equation. The Hamiltonian phase portrait $q(p)$ is a Riemann surface equivalent to reparameterized $ξ$-Newton flow solutions in complex-time, its flow map differential is determined by all Riemann zeros and reminiscent of a 'spectral sum' in trace formulas. Canonical quantization for particle quantum mechanics…

利用开覆盖的全纯对应物的拓扑压力

Topological pressure for holomorphic correspondences using open covers

本文通过开覆盖定义全纯对应物的拓扑压力，并证明其与传统压力定义一致。