On the Correlation Dimension of the Spectral Measure for the Thue-Morse Sequence
نویسندگان
چکیده
Several equivalent definitions are possible; for instance, M ~ = ( 1 ) k+l, where k is the sum of digits in the binary representation ofj. With respect to a shift along the symbolic chain, the Thue-Morse sequence supports a unique translation-invariant ergodic probability measure, t3) Although organized by a simple deterministic rule, the Thue-Morse sequence is not periodic. The Fourier spectrum (in physical applications the term "structure factor" is frequently used) of the infinite chain has long been known to be purely singular continuous: ~4' 5) the spectral measure is neither atomic (discrete) nor absolutely continuous with respect to the Lebesgue measure,
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