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We establish a stability theorem for geometric separating flows on metric spaces, which may or may not be compact. Notably, this result extends Thomas' stability theorem (Proc London Math Soc. 45:479–505 1982). To prove our theorem, we employ set-valued analysis (Aubin and Frankowska, 1990). We give some applications of our result. [ABSTRACT FROM AUTHOR]

Let M be a closed smooth Riemannian manifold and let f : M → M be a diffeomorphism. We show that if f has the C1 robustly asymptotic orbital shadowing property then it is an Anosov diffeomorphism. Moreover, for a C1 generic diffeomorphism f, if f has the asymptotic orbital shadowing property then it is a transitive Anosov diffeomorphism. In particular, we apply our results to volume-preserving diffeomorphisms.

We show that the graph of the classical Weierstrass function ∑n=0∞λncos(2πbnx) has Hausdorff dimension 2+logλ/logb, for every integer b≥2 and every λ∈(1/b,1). Replacing cos(2πx) by a general non-constant C2 periodic function, we obtain the same result under a further assumption that λb is close to 1. [ABSTRACT FROM AUTHOR]

Asymptotically harmonic manifolds are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature $$h$$ . In this article we present results for harmonic functions on rank one asymptotically harmonic manifolds $$X$$ with mild curvature boundedness conditions. Our main results are (a) the explicit calculation of the Radon-Nikodym derivative of the visibility measures, (b) an explicit integral representation for the solution of the Dirichlet problem at infinity in terms of these visibility measures, and (c) a result on horospherical means of bounded eigenfunctions implying that these eigenfunctions do not admit non-trivial continuous extensions to the geometric compactification $$\overline{X}$$ . [ABSTRACT FROM AUTHOR]

We prove that if a local diffeomorphism has expanding periodic points robustly then it is an expanding map. Using this, we reobtain a result due to Sakai: generic positively expansive maps are expanding. Our methods also show a global version of a result by Gan and Yang: generic expansive diffeomorphisms are Axiom A without cycles. [ABSTRACT FROM AUTHOR]