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1. On cusps of caustics by reflection in two dimensional projective Finsler metrics

   https://doi.org/10.2298/TAM250109004T  

   Tabachnikov, Serge (January 2025, Theoretical and Applied Mechanics)

   Given a projective Finsler metric in a convex domain in the projective plane, that is, a metric in which geodesics are straight lines, consider the respective Finsler billiard system. Choose a generic point inside the table and consider the billiard trajectories that start at this point and undergo N reflection off the boundary. The envelope of the resulting 1-parameter family of straight lines is the Nth caustic by reflection. We prove that, for every N, it has at least four cusps, generalizing a similar result for Euclidean metric, obtained recently jointly with G. Bor. 

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2. Outer billiards in the spaces of oriented geodesics of the three‐dimensional space forms

   https://doi.org/10.1112/jlms.12922  

   Godoy, Yamile; Harrison, Michael; Salvai, Marcos (June 2024, Journal of the London Mathematical Society)

   Abstract Let be the three‐dimensional space form of constant curvature , that is, Euclidean space , the sphere , or hyperbolic space . Let be a smooth, closed, strictly convex surface in . We define an outer billiard map on the four‐dimensional space of oriented complete geodesics of , for which the billiard table is the subset of consisting of all oriented geodesics not intersecting . We show that is a diffeomorphism when is quadratically convex. For , has a Kähler structure associated with the Killing form of . We prove that is a symplectomorphism with respect to its fundamental form and that can be obtained as an analogue to the construction of Tabachnikov of the outer billiard in defined in terms of the standard symplectic structure. We show that does not preserve the fundamental symplectic form on associated with the cross product on , for . We initiate the dynamical study of this outer billiard in the hyperbolic case by introducing and discussing a notion of holonomy for periodic points. 

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3. Statistical properties of type D dispersing billiards

   https://doi.org/10.3934/dcds.2022073  

   Brown, Margaret; Nándori, Péter (January 2022, Discrete and Continuous Dynamical Systems)

   We consider dispersing billiard tables whose boundary is piecewise smooth and the free flight function is unbounded. We also assume there are no cusps. Such billiard tables are called type D in the monograph of Chernov and Markarian [9]. For a class of non-degenerate type D dispersing billiards, we prove exponential decay of correlation and several other statistical properties. 

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4. Tooth eruption in the Early Cretaceous British mammal Triconodon and description of a new species

   https://doi.org/10.1002/spp2.1329  

   Jäger, Kai R. K.; Cifelli, Richard L.; Martin, Thomas; Hautier, ed., Lionel (August 2020, Papers in Palaeontology)

   Abstract Triconodon mordax, from the lowest Cretaceous (Berriasian) part of the Purbeck Group, Dorset, is known by an ontogenetic series of specimens that document aspects of tooth eruption and replacement. Based on micro‐computed tomography of four specimens we refer one mandible to a new species,Triconodon averianovi, which differs fromT. mordaxin having a more slender, curved c; p4 notably low crowned with slender main cusp and smaller accessory cusps; and molars with weak cingula, m4 being notably smaller with weak cusps a and c.T. mordaxis variable in the number of mental foramina and posterior jaw morphology. Scans reveal an earlier developmental stage (p3 in early eruption) than previously recognized forTriconodon, and demonstrate sequential, anteroposterior replacement of premolars; it remains unclear whether p1–2 were replaced. Scans also support an earlier hypothesis that m4 erupted late in life. Onset of m4 mineralization is likely to have coincided with eruption of p3, followed by replacement of dp4 by p4 and eruption of c. The m4 developed within the lingual side of the coronoid process, well above the tooth row. It remained in position and was subsequently accommodated in the active tooth row through unusually prolonged and localized growth of the posterior part of the mandible. This pattern is seen in some later triconodontids and appears to be unique to the family. 

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5. Rates of mixing for the measure of maximal entropy of dispersing billiard maps

   https://doi.org/10.1112/plms.12578  

   Demers, Mark F.; Korepanov, Alexey (December 2023, Proceedings of the London Mathematical Society)

   Abstract In a recent work, Baladi and Demers constructed a measure of maximal entropy for finite horizon dispersing billiard maps and proved that it is unique, mixing and moreover Bernoulli. We show that this measure enjoys natural probabilistic properties for Hölder continuous observables, such as at least polynomial decay of correlations and the Central Limit Theorem. The results of Baladi and Demers are subject to a condition of sparse recurrence to singularities. We use a similar and slightly stronger condition, and it has a direct effect on our rate of decay of correlations. For billiard tables with bounded complexity (a property conjectured to be generic), we show that the sparse recurrence condition is always satisfied and the correlations decay at a super‐polynomial rate. 

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