Exact renormalisation equations yield the relative frequency of any patch in a primitive substitution tiling, with transfer to symbolic and other suspension systems.
Complexity and invariant measure of the period-doubling subshift
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abstract
Explicit formulas for complexity and unique invariant measure of the period-doubling subshift can be derived from those for the Thue-Morse subshift, obtained by Brlek, De Luca and Varricchio, and Dekking. In this note we give direct proofs based on combinatorial properties of the period-doubling sequence. We also derive explicit formulas for correlation integral and other recurrence characteristics of the period-doubling subshift. As a corollary we obtain that the determinism of this subshift converges to 1 as the distance threshold approaches 0.
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Exact renormalisation for patch frequencies in inflation systems
Exact renormalisation equations yield the relative frequency of any patch in a primitive substitution tiling, with transfer to symbolic and other suspension systems.