Superconvergence of topological entropy in the symbolic dynamics of substitution sequences

Abstract

We consider infinite sequences of superstable orbits (cascades) generated by systematic substitutions of letters in the symbolic dynamics of one-dimensional nonlinear systems in the logistic map universality class. We identify the conditions under which the topological entropy of successive words converges as a double exponential onto the accumulation point, and find the convergence rates analytically for selected cascades. Numerical tests of the convergence of the control parameter reveal a tendency to quantitatively universal double-exponential convergence. Taking a specific physical example, we consider cascades of stable orbits described by symbolic sequences with the symmetries of quasilattices. We show that all quasilattices can be realised as stable trajectories in nonlinear dynamical systems, extending previous results in which two were identified.

Classical Dimers on Penrose Tilings

“…There are an infinite number in 1D (e.g. the Fibonacci quasilattice [32,57,108]), six in 2D (considered in Section VII), five in 3D (with the point group symmetries of the icosahedron and dodecahedron), one in 4D (with the point group symmetry of the 600-cell), and none in dimension five or higher. All can be generated by inflation rules, matching rules, and by a cut-andproject method from higher dimensions, suggesting that similar methods to those we have developed here may be extended to those cases.…”

“…Any integer power of a PV number is also a PV number. All quasicrystals and Penrose-like tilings can be generated by inflation, and in all cases the largest eigenvalue of the inflation matrix is a quadraticirrational PV number [56,57]. In the thermodynamic limit the largest eigenvalue dominates, and the ratio of the components of the associated right-eigenvector gives the ratio of the number of tiles of each type.…”

“…All known examples of physical quasicrystals have 5-, 8-, 10-, or 12-fold rotational symmetry, and feature symmetries related to quadratic-irrational PV numbers [50,54,57]. In the present paper we have constructed Penrose-like tilings using inflation rules.…”