English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Proof. The penultimate sentence of the proof of [PP, Prop. 4.11] is incorrect. For every n ∈ N, let L1, γn be as in the original version of the proof, and let An be the geodesic line from ∞ to the repelling fixed point γ− n of γn. In order to compute the (strictly increasing) limit, as n tends to +∞, of the approximation constant c(γ− n ) of γ− n (using its expression given by Eq. (11) in [PP]), we need not only to consider the Γ-translates of L1 intersecting An and to minimise 1 − cos θ where θ is its intersection angle, but also to consider the Γ-translates of L1 not intersecting An and to minimise cosh `− 1 where ` is its distance to An. Consider the common perpendicular arc between the translation axis Ln of γn and a disjoint Γ-translate of L1. By the symmetry at i and the computation (done in the original version of the proof) of the translation length of γn, we may restrict to the case when the endpoint on Ln of this common perpendicular arc lies between i and i + n. Let L be the translate by z 7→ z + 1 of L1, whose points at infinity are 3± √ 5 2 . Clearly (see in particular the picture in the original version of the proof), the common perpendicular arc δn between Ln and L realises the minimum distance between Ln and a Γ-translate of L1 disjoint from Ln whose closest point on Ln lies between i and i + n. As n → ∞, the segments δn converge (with strictly increasing lengths) to the common perpendicular arc δ∞ between the positive imaginary axis and L. Since δ∞ is contained in the Euclidean unit circle (which is the angle bisector through i of the equilateral geodesic triangle with vertices i, 1+ i, 1+i 2 ), its hyperbolic length is argcosh 3 √ 5 by an easy computation. Since we analysed the contribution of the Γ-translates of L1 that intersect Ln in the original version of the proof, and since 3 √ 5 − 1 < 1− 1 √ 5 , the (strictly increasing) limit of c(γ− n ) is 3 √ 5 − 1. To conclude, we also need to improve the last claim of the second paragraph of the proof of [PP, Prop. 4.11]. Let T be a triangle as in this second paragraph. The distance from a geodesic line γ meeting T to the geodesic line containing the side of T which is not cut by γ is maximal when γ goes through its opposite vertex and is perpendicular to

Erratum to: Spiraling spectra of geodesic lines in negatively curved manifolds