On the Myrberg Limit Sets and Bowen-Margulis-Sullivan Measures for Visibility Manifolds without Conjugate Points

2024/07/01 by Fei Liu, Liu, Fei · 1 citation
Mathematics · #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2407.00910

openalex publication_date 2024/07/01 · openalex created_date 2024/07/06 · openalex updated_date 2026/07/28

Abstract

In this paper, we clarify the strong relationship between Myrberg type dynamics and the ergodic properties of the geodesic flows on (not necessarily compact) uniform visibility manifolds without conjugate points. We prove that the positivity of the Patterson-Sullivan measure of the Myrberg limit set is equivalent to the conservativity of the geodesic flow with respect to the Bowen-Margulis-Sullivan measure. Moreover we show that the Myrberg limit set is a full Patterson-Sullivan measure subset of the conical limit set. These results extend the classical works of P. Tukia and B. Stratmann from hyperbolic manifolds to the manifolds without conjugate points.

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• On the Dirichlet Problem at Infinity and Poisson Boundary for Certain Manifolds without Conjugate Points

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