Let (Xi, di), i = 1, 2, be proper geodesic hyperbolic metric spaces. We give a general construction for a " hyperbolic product " X1× h X2 which is itself a proper geodesic hyperbolic metric space and examine its boundary at infinity.

Let (M,g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that M is a hyperbolic manifold of constant sectional curvature <EquationSource Format=”TEX”> frac−h24 </EquationSource> , provided M is asymptotically harmonic of constant h > 0. DOI: https://doi.org/10.1007/s00013-008-2611-2 Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-7241 Journal Article Accepted Version Originally published at: Schroeder, Viktor; Shah, H (2008). On 3-dimensional asymptotically harmonic manifolds. Archiv der Mathematik, 90(3):275-278. DOI: https://doi.org/10.1007/s00013-008-2611-2 ar X iv :0 71 0. 07 40 v1 [ m at h. D G ] 3 O ct 2 00 7 On 3-dimensional Asymptotically Harmonic Manifolds Viktor Schroeder & Hemangi Shah

We prove the existence of manifolds with almost maximal volume entropy which are not hyperbolic.

Let \({\mathbf{H}}^n_{{\mathbb K}}\) denote the symmetric space of rank-1 and of non-compact type and let \(d_{{\mathfrak H}}\) be the Korányi metric defined on its boundary. We prove that if d is a metric on \(\partial {\mathbf{H}}^n_{{\mathbb K}}\) such that all Heisenberg similarities are d-Möbius maps, then under a topological condition d is a constant multiple of a power of \(d_{{\mathfrak H}}\).

We define the concept of an ultrametric Möbius space (Z,M) and show that the boundary at infinity of a nonelementary geodesically complete tree is naturally an ultrametric Möbius space. In addition, we construct to a given ultrametric Möbius space (Z,M) a nonelementary geodesically complete tree, unique up to isometry, with (Z,M) being its boundary at infinity. This yields a one-to-one correspondence.

In their earlier work (Ergodic Th. Dynam. Sys., 34: 1699 -1723, 10 2014), the authors introduced the so called F-aperiodic orbits of a dynamical system on a compact metric space X, which satisfy a quantitative condition measuring its recurrence and aperiodicity. Using this condition we introduce two new quantities $\cal{F}$, $\cal{G}$, called the `aperiodic complexities', of the system and establish relations between $\cal{F}$, $\cal{G}$ with the topology and geometry of X. We compare them to well-know complexities such as the box-dimension and the topological entropy. Moreover, we connect our condition to the distribution of periodic orbits and we can classify an F-aperiodic orbit of a point x in X in terms of the collection of the introduced approximation constants of x. Finally, we discuss our results for several examples, in particular for the geodesic flow on hyperbolic manifolds. For each of our examples there is a suitable model of Diophantine approximation and we classify F-aperiodic orbits in terms of Diophantine properties of the point x. As a byproduct, we prove a `metric version' of the closing lemma in the context of CAT(-1) spaces.

We characterize the boundary at infinity of a complex hyperbolic space as a compact Ptolemy space that satisfies four incidence axioms.

A Moebius structure (on a set X) is a class of metrics having the same cross-ratios. A Moebius structure is ptolemaic if it is invariant under inversion operations. The boundary at infinity of a CAT(-1) space is in a natural way a Moebius space, which is ptolemaic. We give a free of classification proof of the following result that characterizes the rank one symmetric spaces of noncompact type purely in terms of their Moebius geometry: Let X be a compact Ptolemy space which contains a Ptolemy circle and allows many space inversions. Then X is Moebius equivalent to the boundary at infinity of a rank one symmetric space.

We introduce a quantitative condition on orbits of dynamical systems, which measures their aperiodicity. We show the existence of sequences in the Bernoulli shift and geodesics on closed hyperbolic manifolds which are as aperiodic as possible with respect to this condition.

In this expository note, we present a transparent proof of Toponogov's theorem for Alexandrov spaces in the general case, not assuming local compactness of the underlying metric space. More precisely, we show that if M is a complete geodesic metric space such that the Alexandrov triangle comparisons for curvature greater than or equal to k are satisfied locally, then these comparisons also hold in the large. The proof is a modification of an argument due to Plaut.

We prove a flat strip theorem for 2-dimensional ptolemaic spaces.

We obtain a Möbius characterization of the n -dimensional spheres S n endowed with the chordal metric d 0 . We show that every compact extended Ptolemy metric space with the property that every three points are contained in a circle is Möbius equivalent to ( S n , d 0 ) for some n ≥ 1.

We characterize the class of Gromov hyperbolic spaces, whose boundary at infinity allow canonical Möbius structures.

In this paper we generalize the Möbius characterization of metric spheres as obtained in Foertsch and Schroeder [4] to a corresponding Möbius characterization of metric hemispheres.

In this paper we characterize Ptolemy circles and Ptolemy segments up to isometry. Moreover, we present an example of a metric sphere, which is Möbius equivalent but not homothetic to the standard metric sphere with its chordal metric.

Using a four points inequality for the boundary of CAT(-1)-spaces we study the relation between Gromov hyperbolic spaces and CAT(-1)-spaces.

In this paper we study geodesic Ptolemy metric spaces X which allow proper and cocompact isometric actions of crystallographic or, more generally, virtual polycyclic groups. We show that X is equivariantly roughly isometric to a Euclidean space.

The paper initiates a systematic study of Moebius structures and Ptolemy spaces. We conjecture that every compact Ptolemy space with circles and many space inversions is Moebius equivalent to the boundary at infinity of a rank one symmetric space of noncompact type. We prove this conjecture for the class of complex hyperbolic spaces as our main result.