Use of the trace map for evaluating localization properties

The use of Lyapunov exponents for evaluating localization lengths of wave f unctions in one-dimensional lattices is discussed. As a result, it is shown that it is more practical to calculate this length by using the scaling pr operties of the trace map of the transfer matrix. This leads to a relations hip between localization and the fixed points of the map, which is consider ed as a dynamical system. The localization length is then defined by a Lyap unov exponent, used in the sense of chaos theory. All these results are dis cussed for periodic, disordered, and quasiperiodic chains. In particular, t he Fibonacci quasiperiodic chain is studied in detail. [S0163-1829(99)05217 -0].