Escape from a circle and Riemann hypotheses

Open circle maps: Small hole asymptotics

We consider escape from chaotic maps through a subset of phase space, the hole. Escape rates are known to be locally constant functions of the hole position and size. In spite of this, for the doubling map we can extend the current best result for small holes, a linear dependence on hole size h, to include a smooth h^2 ln h term and explicit fractal terms to h^2 and higher orders, confirmed by numerical simulations. For more general hole locations the asymptotic form depends on a dynamical Diophantine condition using periodic orbits ordered by stability.

Superstable manifolds of invariant circles and co-dimension 1 Bottcher functions

We consider the situation of a dominant meromorphic self-map $f: X -rightarrow X$, where $X$ is a compact Kähler manifold of dimension $n > 1$. Suppose there is an embedded copy of $\mathbb{P}^1$ that is invariant under $f$, with $f$ holomorphic and transversally superattracting with degree $a$ in some neighborhood. Suppose $f$ restricted to this line is given by $z\mapsto z^b$, with resulting invariant circle $S$. We prove that if $a \geq b$, then the local stable manifold $W^s_\loc(S)$ is real analytic. In fact, we state and prove a suitable localized version that can be useful in wider cont…

内函数、复合算子、符号动力学与热力学形式

Inner Functions, Composition Operators, Symbolic Dynamics and Thermodynamic Formalism

本文利用热力学形式研究单位圆盘内函数的动力学，通过复合算子等联系建立热力学形式证明相关定理，在单分量内函数假设下实现轨道计数，还讨论了单位圆上Denjoy-Wolff点的内函数并推导随机律等结果。