Multidimensional contracted rotations

2025/09/22 by José Pedro Gaivão, Gaivao, Jose Pedro, Benito Pires +1
Engineering · #Dynamical Systems (math.DS) #FOS: Mathematics #Geophysics and Sensor Technology

paper · pdf · doi:10.48550/arxiv.2509.17639

openalex publication_date 2025/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We study the dynamics of multidimensional contracted rotations and address a problem posed by Y. Bugeaud and J-P. Conze in Acta Arithmetica in 1999. More precisely, we show that if A is an invertible linear contraction of ℝ^d, then the map f: [0,1)^d→ [0,1)^d defined by f(x) = Ax +b (\textrmmod ℤ^d) is asymptotically periodic for Lebesgue almost all b∈ℝ^d. We also include an example of a family of multidimensional contracted rotations (d>1) not conjugate to the product of one-dimensional contracted rotations (d=1), showing that our result cannot be reduced to or derived from the one-dimensional result of Bugeaud and Conze.

Citations

• Multi-dimensional piecewise contractions are asymptotically periodic
• On the Diophantine nature of the elements of Cantor sets arising in the dynamics of contracted rotations
• On the Asymptotic Properties of Piecewise Contracting Maps
• Discrete time piecewise affine models of genetic regulatory networks

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