On the normal exponential map in singular conformal metrics

Non-Escaping Sets in Conformal Dynamical Systems and Singular Perturbations of Perron-Frobenius Operators

The study of escape rates for a ball in a dynamical systems has been much studied. Understanding the asymptotic behavior of the escape rate as the radius of the ball tends to zero is an especially subtle problem. In the case of hyperbolic conformal systems this has been addressed by various authors. In this paper we consider a far more general realm of conformal maps where the analysis is correspondingly more complicated. We prove the existence of escape rates and calculate them in the context of countable alphabets, either finite or infinite, uniformly contracting conformal graph directed Mar…

共形通用子黎曼度量的正则奇异测地线

Normal Singular Geodesics of a Conformally Generic Sub-Riemannian Metric

该研究证明了在特定分布下，即使加入势函数，也不存在非平凡的正则奇异轨道。

Repelling periodic points and landing of rays for post-singularly bounded exponential maps

We show that repelling periodic points are landing points of periodic rays for exponential maps whose singular value has bounded orbit. For polynomials with connected Julia sets, this is a celebrated theorem by Douady, for which we present a new proof. In both cases we also show that points in hyperbolic sets are accessible by at least one and at most finitely many rays. For exponentials this allows us to conclude that the singular value itself is accessible.

Directed porosity on conformal iterated function systems and weak convergence of singular integrals

The aim of the present paper is twofold. We study directed porosity in connection with conformal iterated function systems (CIFS) and with singular integrals. We prove that limit sets of finite CIFS are porous in a stronger sense than already known. Furthermore we use directed porosity to establish that truncated singular integral operators, with respect to general Radon measures $μ$ and kernels $K$, converge weakly in some dense subspaces of $L^2(μ)$ when the support of $μ$ belongs to a broad family of sets. This class contains many fractal sets like CIFS's limit sets.

Singularly Perturbed Oscillators with Exponential Nonlinearities

Singular exponential nonlinearities of the form $e^{h(x)ε^{-1}}$ with $ε>0$ small occur in many different applications. These terms have essential singularities for $ε=0$ leading to very different behaviour depending on the sign of $h$. In this paper, we consider two prototypical singularly perturbed oscillators with such exponential nonlinearities. We apply a suitable normalization for both systems such that the $ε\rightarrow 0$ limit is a piecewise smooth system. The convergence to this nonsmooth system is exponential due to the nonlinearities we study. By working on the two model systems we…

Morse Theory for geodesics in singular conformal metrics

Motivated by the use of degenerate Jacobi metrics for the study of brake orbits and homoclinics, we develop a Morse theory for geodesics in conformal metrics having conformal factors vanishing on a regular hypersurface of a Riemannian manifold.