Trace maps, invariants, and some of their applications

Abstract

Trace maps of two-letter substitution rules are investigated with special emphasis on the underlying algebraic structure and on the existence of invariants. We illustrate the results with the generalized Fibonacci chains and show that the well-known Fricke character I(x,y,z) = x^2 + y^2 + z^2 - 2 x y z - 1 is not the only type of invariant that can occur. We discuss several physical applications to electronic spectra including the gap-labeling theorem, to kicked two-level systems, and to the classical 1D Ising model with non-commuting transfer matrices.

23 pages, including 2 figures, paper made available here due to renewed interest

Cited by in corpus (24)

• Gordon-type arguments in the spectral theory of one-dimensional quasicrystals
• Trace and antitrace maps for aperiodic sequences, their extensions and applications
• Energy spectra, wavefunctions and quantum diffusion for quasiperiodic systems
• Spectral and Quantum Dynamical Properties of the Weakly Coupled Fibonacci Hamiltonian
• Hyperbolicity of the Trace Map for the Weakly Coupled Fibonacci Hamiltonian
• The fine structure of heating in a quasiperiodically driven critical quantum system
• A note on palindromicity
• Spectral and Diffusive Properties of Silver-Mean Quasicrystals in 1,2, and 3 Dimensions
• Renormalization Group Approach for the Wave Packet Dynamics in Golden-Mean and Silver-Mean Labyrinth Tilings
• Spectral analysis of tridiagonal Fibonacci Hamiltonians
• Local symmetry theory of resonator structures for the real-space control of edge states in binary aperiodic chains
• Symmetries and reversing symmetries of toral automorphisms
• Energy spectra and eigenstates of quasiperiodic tight-binding Hamiltonians
• Anomalous electronic conductance in quasicrystals
• Properties of 1D classical and quantum Ising quasicrystals: rigorous results
• Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices
• Driven Non-Unitary Dynamics of Quantum Critical Systems
• The Fibonacci quasicrystal: case study of hidden dimensions and multifractality
• On the spectrum of 1D quantum Ising quasicrystal
• On the entropy and letter frequencies of powerfree words
• Absolutely continuous energy bands and extended electronic states in an aperiodic comb-shaped nanostructure
• Reversing Symmetry Groups of Cat Maps
• Points with finite orbits for trace maps
• Newhouse phenomena in the Fibonacci trace map