2024/03/29 by Joseph A. Beck, W. W. L. Chen, Beck, J. +3
Engineering · Mathematics · #11K38 #37E35 #Advanced Materials and Mechanics #Architecture and Computational Design #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2403.19954
openalex publication_date 2024/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The class of 2-dimensional non-integrable flat dynamical systems has a rather extensive literature with many deep results, but the methods developed for this type of problems, both the traditional approach via Teichmüller geometry and our recent shortline-ancestor method, appear to be exclusively plane-specific. Thus we know very little of any real significance concerning 3-dimensional systems. Our purpose here is to describe some very limited extensions of uniformity in 2 dimensions to uniformity in 3 dimensions. We consider a 3-manifold which is the cartesian product of the regular octagonal surface with the unit torus. This is a restricted system, in the sense that one of the directions is integrable. However, this restriction also allows us to make use of a transference theorem for arithmetic progressions established earlier by Beck, Donders and Yang.