2010/01/29 by Jeong-Yup Lee, Lee, Jeong-Yup, Boris Solomyak +1
Computer Science · Mathematics · #37B50 #52C23 #Cellular Automata and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Nonlinear Dynamics and Pattern Formation #math.DS #math.MG #msc:37B50 #msc:52C23
paper · pdf · doi:10.48550/arxiv.1002.0039
26 pages
openalex publication_date 2010/01/29 · arxiv created 2010/01/30 · arxiv updated 2010/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider self-affine tilings in the Euclidean space and the associated tiling dynamical systems, namely, the translation action on the orbit closure of the given tiling. We investigate the spectral properties of the system. It turns out that the presence of the discrete component depends on the algebraic properties of the eigenvalues of the expansion matrix ϕ for the tiling. Assuming that ϕ is diagonalizable over \C and all its eigenvalues are algebraic conjugates of the same multiplicity, we show that the dynamical system has a relatively dense discrete spectrum if and only if it is not weakly mixing, and if and only if the spectrum of ϕ is a "Pisot family". Moreover, this is equivalent to the Meyer property of the associated discrete set of "control points" for the tiling.