partition function for a smaller system with but spins, provided we replace and with renormalized values and satisfying the equations Suppose now that we allow to be complex and that for some . In that case we can introduce a parameter related to via the equation and the resulting renormalization group transformation for will be precisely the dyadic map: See also Bernoulli process Bernoulli scheme Gilbert–Shannon–Reeds model, a random distribution on permutations given by applying the doubling map to a set of n uniformly random points on the unit interval Notes References Dean J. Driebe, Fully Chaotic Maps and Broken Time Symmetry, (1999) Kluwer Academic Publishers, Dordrecht Netherlands Linas Vepstas, The Bernoulli Map, the Gauss-Kuzmin-Wirsing Operator and the Riemann Zeta, (2004) Chaotic maps