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Number of scaling factors in incommensurate systems

The authors study the scaling properties of an incommensurate system which obeys simple clustering rules, the one-dimensional tight-binding Schrodinger equation with a Fibonacci sequence potential. They find that the clustering rules do not guarantee a finite number of scaling factors. They find the energy spectrum by recursion. The clustering rules and recursion relations suggest different descriptions of the same spectrum. Based on the thermodynamic formalism, they compare these descriptions, calculate their associated scaling functions, and construct the mapping between them.