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We review the recent developments in the theory of the one-dimensionaltight-binding Schr\"odinger equation for a class of deterministic ergodicpotentials. In the typical examples the potentials are generated bysubstitutional sequences, like the Fibonacci or the Thue-Morse sequence. Weconcentrate on rigorous results which will be explained rather than proved. Thenecessary mathematical background is provided in the text.

Schroedinger difference equation with deterministic ergodic potentials